X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_nstdsums.cpp;h=84a78c42c8e45d6c7b2ee76a9475fd42922f59d5;hp=020ff333b01399329df89dd03e5fc34490087992;hb=9d92d4b442fc4c1a95685884be4ba0494cd02bbe;hpb=168dd38dd35fcaaf61e177afb8dbe0288c0eb521 diff --git a/ginac/inifcns_nstdsums.cpp b/ginac/inifcns_nstdsums.cpp index 020ff333..84a78c42 100644 --- a/ginac/inifcns_nstdsums.cpp +++ b/ginac/inifcns_nstdsums.cpp @@ -1,52 +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 + * representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only * 0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single - * number --- notation. - * - * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in - * the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product - * x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers. - * If you want to have an alternating Euler sum, you have to give the signs of the parameters as a - * second argument s to zeta(m,s) containing 1 and -1. - * - * - The calculation of classical polylogarithms is speed up by using Bernoulli numbers and + * 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 calculation of classical polylogarithms is speeded up by using Bernoulli numbers and * look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in - * [Cra] and [BBB] for speed up. - * - * - The functions have no series expansion into nested sums. To do this, you have to convert these functions - * into the appropriate objects from the nestedsums library, do the expansion and convert the - * result back. - * + * [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) 1999-2008 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 @@ -60,9 +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 @@ -102,6 +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; @@ -117,17 +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 { @@ -147,14 +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; @@ -171,9 +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++; } @@ -183,13 +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 { @@ -207,17 +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; } @@ -226,7 +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; @@ -244,8 +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; @@ -253,15 +308,19 @@ 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; } // forward declaration needed by function Li_projection and C below -numeric S_num(int n, int p, const numeric& x); +const cln::cl_N S_num(int n, int p, const cln::cl_N& x); // helper function for classical polylog Li @@ -312,23 +371,22 @@ cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& pr } 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(cln::realpart(value))); - else if (!x.imag().is_rational()) + else if (!instanceof(imagpart(x), cln::cl_RA_ring)) prec = cln::float_format(cln::the(cln::imagpart(value))); // [Kol] (5.15) @@ -373,7 +439,7 @@ numeric Li_num(int n, const numeric& x) cln::cl_N add; for (int j=0; j& s, const std::vector& x) { + // ensure all x <> 0. + for (std::vector::const_iterator it = x.begin(); it != x.end(); ++it) { + if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits)); + } + const int j = s.size(); + bool flag_accidental_zero = false; std::vector t(j); cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); @@ -411,111 +484,1188 @@ 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]); } - // ... 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--) { + flag_accidental_zero = cln::zerop(t[k+1]); t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]); } - } while (t[0] != t0buf); + } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero ); return t[0]; } +// forward declaration for Li_eval() +lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf); + + +// type used by the transformation functions for G +typedef std::vector Gparameter; + + +// G_eval1-function for G transformations +ex G_eval1(int a, int scale, const exvector& gsyms) +{ + if (a != 0) { + const ex& scs = gsyms[std::abs(scale)]; + const ex& as = gsyms[std::abs(a)]; + if (as != scs) { + return -log(1 - scs/as); + } else { + return -zeta(1); + } + } else { + return log(gsyms[std::abs(scale)]); + } +} + + +// G_eval-function for G transformations +ex G_eval(const Gparameter& a, int scale, const exvector& gsyms) +{ + // check for properties of G + ex sc = gsyms[std::abs(scale)]; + lst newa; + bool all_zero = true; + bool all_ones = true; + int count_ones = 0; + for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) { + if (*it != 0) { + const ex sym = gsyms[std::abs(*it)]; + newa.append(sym); + all_zero = false; + if (sym != sc) { + all_ones = false; + } + if (all_ones) { + ++count_ones; + } + } else { + all_ones = false; + } + } + + // 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, gsyms) * G_eval(short_a, scale, gsyms); + it = short_a.begin(); + for (int i=1; i G({1};y)^k / k! + if (all_ones && a.size() > 1) { + return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones); + } + + // G({0,...,0};y) -> log(y)^k / k! + if (all_zero) { + return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size()); + } + + // no special cases anymore -> convert it into Li + lst m; + lst x; + ex argbuf = gsyms[std::abs(scale)]; + ex mval = _ex1; + for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) { + if (*it != 0) { + const ex& sym = gsyms[std::abs(*it)]; + x.append(argbuf / sym); + m.append(mval); + mval = _ex1; + argbuf = sym; + } else { + ++mval; + } + } + return pow(-1, x.nops()) * Li(m, x); +} + + +// converts data for G: pending_integrals -> a +Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals) +{ + 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 { + // just return empty parameter list + Gparameter new_a; + return new_a; + } +} + + +// 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; +} + + +// 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); + + if (pending_integrals.size() > 0) { + return pending_integrals; + } else { + Gparameter new_pending_integrals; + new_pending_integrals.push_back(scale); + return new_pending_integrals; + } +} + + +// handles trailing zeroes for an otherwise convergent integral +ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms) +{ + bool convergent; + int depth, trailing_zeros; + Gparameter::const_iterator last, dummyit; + last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit); + + GINAC_ASSERT(convergent); + + if ((trailing_zeros > 0) && (depth > 0)) { + ex result; + Gparameter new_a(a.begin(), a.end()-1); + result += G_eval1(0, scale, gsyms) * trailing_zeros_G(new_a, scale, gsyms); + for (Gparameter::const_iterator it = a.begin(); it != last; ++it) { + Gparameter new_a(a.begin(), it); + new_a.push_back(0); + new_a.insert(new_a.end(), it, a.end()-1); + result -= trailing_zeros_G(new_a, scale, gsyms); + } + + return result / trailing_zeros; + } else { + return G_eval(a, scale, gsyms); + } +} + + +// G transformation [VSW] (57),(58) +ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale, const exvector& gsyms) +{ + // 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(), + gsyms); + } + + // G(y2_{-+}; sr) + result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), + new_pending_integrals.front(), + gsyms); + + // G(0; sr) + new_pending_integrals.back() = 0; + result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), + new_pending_integrals.front(), + gsyms); + + return result; + } + + // length > 1 + // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t ) + // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t ) + + //term zeta_m + result -= zeta(a.size()); + if (psize) { + result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), + pending_integrals.front(), + gsyms); + } + + // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t ) + // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 ) + Gparameter new_a(a.begin()+1, a.end()); + new_pending_integrals.push_back(0); + result -= depth_one_trafo_G(new_pending_integrals, new_a, scale, gsyms); + + // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t ) + // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 ) + Gparameter new_pending_integrals_2; + new_pending_integrals_2.push_back(scale); + new_pending_integrals_2.push_back(0); + if (psize) { + result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals), + pending_integrals.front(), + gsyms) + * depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms); + } else { + result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms); + } + + return result; +} + + +// forward declaration +ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2, + const Gparameter& pendint, const Gparameter& a_old, int scale, + const exvector& gsyms); + + +// G transformation [VSW] +ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale, + const exvector& gsyms) +{ + // main recursion routine + // + // pendint = ( y1, b1, ..., br ) + // a = ( a1, ..., amin, ..., aw ) + // scale = y2 + // + // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2) + // where sr replaces amin + + // find smallest alpha, determine depth and trailing zeros, and check for convergence + bool convergent; + int depth, trailing_zeros; + Gparameter::const_iterator min_it; + Gparameter::const_iterator firstzero = + check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it); + int min_it_pos = min_it - a.begin(); + + // special case: all a's are zero + if (depth == 0) { + ex result; + + if (a.size() == 0) { + result = 1; + } else { + result = G_eval(a, scale, gsyms); + } + if (pendint.size() > 0) { + result *= trailing_zeros_G(convert_pending_integrals_G(pendint), + pendint.front(), + gsyms); + } + return result; + } + + // handle trailing zeros + if (trailing_zeros > 0) { + ex result; + Gparameter new_a(a.begin(), a.end()-1); + result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms); + for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) { + Gparameter new_a(a.begin(), it); + new_a.push_back(0); + new_a.insert(new_a.end(), it, a.end()-1); + result -= G_transform(pendint, new_a, scale, gsyms); + } + return result / trailing_zeros; + } + + // convergence case + if (convergent) { + if (pendint.size() > 0) { + return G_eval(convert_pending_integrals_G(pendint), + pendint.front(), gsyms)* + G_eval(a, scale, gsyms); + } else { + return G_eval(a, scale, gsyms); + } + } + + // call basic transformation for depth equal one + if (depth == 1) { + return depth_one_trafo_G(pendint, a, scale, gsyms); + } + + // do recursion + // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2) + // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2) + // + int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) int_0^{sr} ds_{r+1} d/ds_{r+1} G(a1,...,s_{r+1},...,aw,y2) + + // smallest element in last place + if (min_it + 1 == a.end()) { + do { --min_it; } while (*min_it == 0); + Gparameter empty; + Gparameter a1(a.begin(),min_it+1); + Gparameter a2(min_it+1,a.end()); + + ex result = G_transform(pendint, a2, scale, gsyms)* + G_transform(empty, a1, scale, gsyms); + + result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms); + return result; + } + + Gparameter empty; + Gparameter::iterator changeit; + + // first term G(a_1,..,0,...,a_w;a_0) + Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]); + Gparameter new_a = a; + new_a[min_it_pos] = 0; + ex result = G_transform(empty, new_a, scale, gsyms); + if (pendint.size() > 0) { + result *= trailing_zeros_G(convert_pending_integrals_G(pendint), + pendint.front(), gsyms); + } + + // other terms + changeit = new_a.begin() + min_it_pos; + changeit = new_a.erase(changeit); + if (changeit != new_a.begin()) { + // smallest in the middle + new_pendint.push_back(*changeit); + result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms); + int buffer = *changeit; + *changeit = *min_it; + result += G_transform(new_pendint, new_a, scale, gsyms); + *changeit = buffer; + new_pendint.pop_back(); + --changeit; + new_pendint.push_back(*changeit); + result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms); + *changeit = *min_it; + result -= G_transform(new_pendint, new_a, scale, gsyms); + } else { + // smallest at the front + new_pendint.push_back(scale); + result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms); + new_pendint.back() = *changeit; + result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms); + *changeit = *min_it; + result += G_transform(new_pendint, new_a, scale, gsyms); + } + return result; +} + + +// shuffles the two parameter list a1 and a2 and calls G_transform for every term except +// for the one that is equal to a_old +ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2, + const Gparameter& pendint, const Gparameter& a_old, int scale, + const exvector& gsyms) +{ + if (a1.size()==0 && a2.size()==0) { + // veto the one configuration we don't want + if ( a0 == a_old ) return 0; + + return G_transform(pendint, a0, scale, gsyms); + } + + if (a2.size()==0) { + Gparameter empty; + Gparameter aa0 = a0; + aa0.insert(aa0.end(),a1.begin(),a1.end()); + return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms); + } + + if (a1.size()==0) { + Gparameter empty; + Gparameter aa0 = a0; + aa0.insert(aa0.end(),a2.begin(),a2.end()); + return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms); + } + + Gparameter a1_removed(a1.begin()+1,a1.end()); + Gparameter a2_removed(a2.begin()+1,a2.end()); + + Gparameter a01 = a0; + Gparameter a02 = a0; + + a01.push_back( a1[0] ); + a02.push_back( a2[0] ); + + return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms) + + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms); +} + +// handles the transformations and the numerical evaluation of G +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_numeric(const std::vector& x, const std::vector& s, + const cln::cl_N& y); + +// do acceleration transformation (hoelder convolution [BBB]) +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_do_hoelder(std::vector x, /* yes, it's passed by value */ + const std::vector& s, const cln::cl_N& y) +{ + cln::cl_N result; + const std::size_t size = x.size(); + for (std::size_t i = 0; i < size; ++i) + x[i] = x[i]/y; + + for (std::size_t r = 0; r <= size; ++r) { + cln::cl_N buffer(1 & r ? -1 : 1); + cln::cl_RA p(2); + bool adjustp; + do { + adjustp = false; + for (std::size_t i = 0; i < size; ++i) { + if (x[i] == cln::cl_RA(1)/p) { + p = p/2 + cln::cl_RA(3)/2; + adjustp = true; + continue; + } + } + } while (adjustp); + cln::cl_RA q = p/(p-1); + std::vector qlstx; + std::vector qlsts; + for (std::size_t j = r; j >= 1; --j) { + qlstx.push_back(cln::cl_N(1) - x[j-1]); + if (instanceof(x[j-1], cln::cl_R_ring) && + realpart(x[j-1]) > 1 && realpart(x[j-1]) <= 2) { + qlsts.push_back(s[j-1]); + } else { + qlsts.push_back(-s[j-1]); + } + } + if (qlstx.size() > 0) { + buffer = buffer*G_numeric(qlstx, qlsts, 1/q); + } + std::vector plstx; + std::vector plsts; + for (std::size_t j = r+1; j <= size; ++j) { + plstx.push_back(x[j-1]); + plsts.push_back(s[j-1]); + } + if (plstx.size() > 0) { + buffer = buffer*G_numeric(plstx, plsts, 1/p); + } + result = result + buffer; + } + return result; +} + +// convergence transformation, used for numerical evaluation of G function. +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_do_trafo(const std::vector& x, const std::vector& s, + const cln::cl_N& y) +{ + // sort (|x|<->position) to determine indices + typedef std::multimap sortmap_t; + sortmap_t sortmap; + std::size_t size = 0; + for (std::size_t i = 0; i < x.size(); ++i) { + if (!zerop(x[i])) { + sortmap.insert(std::make_pair(abs(x[i]), i)); + ++size; + } + } + // include upper limit (scale) + sortmap.insert(std::make_pair(abs(y), x.size())); + + // generate missing dummy-symbols + int i = 1; + // holding dummy-symbols for the G/Li transformations + exvector gsyms; + gsyms.push_back(symbol("GSYMS_ERROR")); + cln::cl_N lastentry(0); + for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) { + if (it != sortmap.begin()) { + if (it->second < x.size()) { + if (x[it->second] == lastentry) { + gsyms.push_back(gsyms.back()); + continue; + } + } else { + if (y == lastentry) { + gsyms.push_back(gsyms.back()); + continue; + } + } + } + std::ostringstream os; + os << "a" << i; + gsyms.push_back(symbol(os.str())); + ++i; + if (it->second < x.size()) { + lastentry = x[it->second]; + } else { + lastentry = y; + } + } + + // fill position data according to sorted indices and prepare substitution list + Gparameter a(x.size()); + exmap subslst; + std::size_t pos = 1; + int scale; + for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) { + if (it->second < x.size()) { + if (s[it->second] > 0) { + a[it->second] = pos; + } else { + a[it->second] = -int(pos); + } + subslst[gsyms[pos]] = numeric(x[it->second]); + } else { + scale = pos; + subslst[gsyms[pos]] = numeric(y); + } + ++pos; + } + + // do transformation + Gparameter pendint; + ex result = G_transform(pendint, a, scale, gsyms); + // replace dummy symbols with their values + result = result.eval().expand(); + result = result.subs(subslst).evalf(); + if (!is_a(result)) + throw std::logic_error("G_do_trafo: G_transform returned non-numeric result"); + + cln::cl_N ret = ex_to(result).to_cl_N(); + return ret; +} + +// handles the transformations and the numerical evaluation of G +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_numeric(const std::vector& x, const std::vector& s, + const cln::cl_N& y) +{ + // check for convergence and necessary accelerations + bool need_trafo = false; + bool need_hoelder = false; + std::size_t depth = 0; + for (std::size_t i = 0; i < x.size(); ++i) { + if (!zerop(x[i])) { + ++depth; + const cln::cl_N x_y = abs(x[i]) - y; + if (instanceof(x_y, cln::cl_R_ring) && + realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2))) + need_trafo = true; + + if (abs(abs(x[i]/y) - 1) < 0.01) + need_hoelder = true; + } + } + if (zerop(x[x.size() - 1])) + need_trafo = true; + + if (depth == 1 && x.size() == 2 && !need_trafo) + return - Li_projection(2, y/x[1], cln::float_format(Digits)); + + // do acceleration transformation (hoelder convolution [BBB]) + if (need_hoelder) + return G_do_hoelder(x, s, y); + + // convergence transformation + if (need_trafo) + return G_do_trafo(x, s, y); + + // do summation + std::vector newx; + newx.reserve(x.size()); + std::vector m; + m.reserve(x.size()); + int mcount = 1; + int sign = 1; + cln::cl_N factor = y; + for (std::size_t i = 0; i < x.size(); ++i) { + if (zerop(x[i])) { + ++mcount; + } else { + newx.push_back(factor/x[i]); + factor = x[i]; + m.push_back(mcount); + mcount = 1; + sign = -sign; + } + } + + return sign*multipleLi_do_sum(m, newx); +} + + +ex mLi_numeric(const lst& m, const lst& x) +{ + // let G_numeric do the transformation + std::vector newx; + newx.reserve(x.nops()); + std::vector s; + s.reserve(x.nops()); + cln::cl_N factor(1); + for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { + for (int i = 1; i < *itm; ++i) { + newx.push_back(cln::cl_N(0)); + s.push_back(1); + } + const cln::cl_N xi = ex_to(*itx).to_cl_N(); + newx.push_back(factor/xi); + factor = factor/xi; + s.push_back(1); + } + return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1))); +} + + } // end of anonymous namespace ////////////////////////////////////////////////////////////////////// // -// Classical polylogarithm and multiple polylogarithm Li(n,x) +// Generalized multiple polylogarithm G(x, y) and G(x, s, y) // // GiNaC function // ////////////////////////////////////////////////////////////////////// -static ex Li_evalf(const ex& x1, const ex& x2) +static ex G2_evalf(const ex& x_, const ex& y) { - // classical polylogs - if (is_a(x1) && is_a(x2)) { - return Li_num(ex_to(x1).to_int(), ex_to(x2)); + if (!y.info(info_flags::positive)) { + return G(x_, y).hold(); } - // multiple polylogs - else if (is_a(x1) && is_a(x2)) { - ex conv = 1; - for (int i=0; i(x_) ? ex_to(x_) : lst(x_); + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, y).hold(); + } + std::vector s; + s.reserve(x.nops()); + bool all_zero = true; + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) { + if (!(*it).info(info_flags::numeric)) { + return G(x_, y).hold(); + } + if (*it != _ex0) { + all_zero = false; + } + if ( !ex_to(*it).is_real() && ex_to(*it).imag() < 0 ) { + s.push_back(-1); + } + else { + s.push_back(1); + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + std::vector xv; + xv.reserve(x.nops()); + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) + xv.push_back(ex_to(*it).to_cl_N()); + cln::cl_N result = G_numeric(xv, s, ex_to(y).to_cl_N()); + return numeric(result); +} + + +static ex G2_eval(const ex& x_, const ex& y) +{ + //TODO eval to MZV or H or S or Lin + + if (!y.info(info_flags::positive)) { + return G(x_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst(x_); + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, y).hold(); + } + std::vector s; + s.reserve(x.nops()); + bool all_zero = true; + bool crational = true; + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) { + if (!(*it).info(info_flags::numeric)) { + return G(x_, y).hold(); + } + if (!(*it).info(info_flags::crational)) { + crational = false; + } + if (*it != _ex0) { + all_zero = false; + } + if ( !ex_to(*it).is_real() && ex_to(*it).imag() < 0 ) { + s.push_back(-1); + } + else { + s.push_back(+1); + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + if (!y.info(info_flags::crational)) { + crational = false; + } + if (crational) { + return G(x_, y).hold(); + } + std::vector xv; + xv.reserve(x.nops()); + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) + xv.push_back(ex_to(*it).to_cl_N()); + cln::cl_N result = G_numeric(xv, s, ex_to(y).to_cl_N()); + return numeric(result); +} + + +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(); + } + std::vector sn; + sn.reserve(s.nops()); + bool all_zero = true; + for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) { + if (!(*itx).info(info_flags::numeric)) { + return G(x_, y).hold(); + } + if (!(*its).info(info_flags::real)) { + return G(x_, y).hold(); + } + if (*itx != _ex0) { + all_zero = false; + } + if ( ex_to(*itx).is_real() ) { + if ( *its >= 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } + else { + if ( ex_to(*itx).imag() > 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + std::vector xn; + xn.reserve(x.nops()); + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) + xn.push_back(ex_to(*it).to_cl_N()); + cln::cl_N result = G_numeric(xn, sn, ex_to(y).to_cl_N()); + return numeric(result); +} + + +static ex G3_eval(const ex& x_, const ex& s_, const ex& y) +{ + //TODO eval to MZV or H or S or Lin + + if (!y.info(info_flags::positive)) { + return G(x_, s_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst(x_); + lst s = is_a(s_) ? ex_to(s_) : lst(s_); + if (x.nops() != s.nops()) { + return G(x_, s_, y).hold(); + } + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, s_, y).hold(); + } + std::vector sn; + sn.reserve(s.nops()); + bool all_zero = true; + bool crational = true; + for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) { + if (!(*itx).info(info_flags::numeric)) { + return G(x_, s_, y).hold(); + } + if (!(*its).info(info_flags::real)) { + return G(x_, s_, y).hold(); + } + if (!(*itx).info(info_flags::crational)) { + crational = false; + } + if (*itx != _ex0) { + all_zero = false; + } + if ( ex_to(*itx).is_real() ) { + if ( *its >= 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } + else { + if ( ex_to(*itx).imag() > 0 ) { + sn.push_back(1); } - if (!is_a(x2.op(i))) { - return Li(x1,x2).hold(); + else { + sn.push_back(-1); } - conv *= x2.op(i); - if ((conv > 1) || ((conv == 1) && (x1.op(0) == 1))) { - return Li(x1,x2).hold(); + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + if (!y.info(info_flags::crational)) { + crational = false; + } + if (crational) { + return G(x_, s_, y).hold(); + } + std::vector xn; + xn.reserve(x.nops()); + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) + xn.push_back(ex_to(*it).to_cl_N()); + cln::cl_N result = G_numeric(xn, sn, ex_to(y).to_cl_N()); + return numeric(result); +} + + +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)) { + int m__ = ex_to(m_).to_int(); + const cln::cl_N x__ = ex_to(x_).to_cl_N(); + const cln::cl_N result = Lin_numeric(m__, x__); + return numeric(result); + } else { + // try to numerically evaluate second argument + ex x_val = x_.evalf(); + if (x_val.info(info_flags::numeric)) { + int m__ = ex_to(m_).to_int(); + const cln::cl_N x__ = ex_to(x_val).to_cl_N(); + const cln::cl_N result = Lin_numeric(m__, x__); + return numeric(result); } } + } + // multiple polylogs + if (is_a(m_) && is_a(x_)) { - 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()); + 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(); } - return numeric(multipleLi_do_sum(m, x)); + 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(x1,x2).hold(); + return Li(m_,x_).hold(); } -static ex Li_eval(const ex& x1, const ex& x2) +static ex Li_eval(const ex& m_, const ex& x_) { - if (x2.is_zero()) { - return _ex0; - } - else { - if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) - return Li_num(ex_to(x1).to_int(), ex_to(x2)); - if (is_a(x2)) { - for (int i=0; i(x2.op(i))) { - return Li(x1,x2).hold(); + if (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; } } - return Li(x1,x2).evalf(); + 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; } - return Li(x1,x2).hold(); } + if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) { + int m__ = ex_to(m_).to_int(); + const cln::cl_N x__ = ex_to(x_).to_cl_N(); + const cln::cl_N result = Lin_numeric(m__, x__); + return numeric(result); + } + + return Li(m_, x_).hold(); } -static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options) +static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options) { - epvector seq; - seq.push_back(expair(Li(x1,x2), 0)); - return pseries(rel,seq); + if (is_a(m) || is_a(x)) { + // multiple polylog + epvector seq; + seq.push_back(expair(Li(m, x), 0)); + return pseries(rel, seq); + } + + // classical polylog + const ex x_pt = x.subs(rel, subs_options::no_pattern); + if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) { + // First special case: x==0 (derivatives have poles) + if (x_pt.is_zero()) { + const symbol s; + ex ser; + // manually construct the primitive expansion + for (int i=1; i=1 (branch cut) + throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!"); + } + // all other cases should be safe, by now: + throw do_taylor(); // caught by function::series() } -static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param) +static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param) { GINAC_ASSERT(deriv_param < 2); if (deriv_param == 0) { return _ex0; } - if (x1 > 0) { - return Li(x1-1, x2) / x2; + if (m_.nops() > 1) { + throw std::runtime_error("don't know how to derivate multiple polylogarithm!"); + } + ex m; + if (is_a(m_)) { + m = m_.op(0); + } else { + m = m_; + } + ex x; + if (is_a(x_)) { + x = x_.op(0); } else { - return 1/(1-x2); + x = x_; + } + if (m > 0) { + return Li(m-1, x) / x; + } else { + return 1/(1-x); } } @@ -555,12 +1705,12 @@ static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c) REGISTER_FUNCTION(Li, - evalf_func(Li_evalf). - eval_func(Li_eval). - series_func(Li_series). - derivative_func(Li_deriv). - print_func(Li_print_latex). - do_not_evalf_params()); + evalf_func(Li_evalf). + eval_func(Li_eval). + series_func(Li_series). + derivative_func(Li_deriv). + print_func(Li_print_latex). + do_not_evalf_params()); ////////////////////////////////////////////////////////////////////// @@ -669,10 +1819,10 @@ cln::cl_N C(int n, int p) if (k == 0) { if (n & 1) { if (j & 1) { - result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j); + result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j); } else { - result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j); + result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j); } } } @@ -680,24 +1830,24 @@ cln::cl_N C(int n, int p) if (k & 1) { if (j & 1) { result = result + cln::factorial(n+k-1) - * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() - / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); + * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1) + / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); } else { result = result - cln::factorial(n+k-1) - * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() - / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); + * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1) + / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); } } else { if (j & 1) { - result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() - / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); + result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1) + / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); } else { result = result + cln::factorial(n+k-1) - * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() - / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); + * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1) + / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); } } } @@ -758,10 +1908,20 @@ cln::cl_N b_k(int k) // helper function for S(n,p,x) cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec) { + static cln::float_format_t oldprec = cln::default_float_format; + if (p==1) { return Li_projection(n+1, x, prec); } - + + // precision has changed, we need to clear lookup table Yn + if ( oldprec != prec ) { + Yn.clear(); + ynsize = 0; + ynlength = 100; + oldprec = prec; + } + // check if precalculated values are sufficient if (p > ynsize+1) { for (int i=ynsize; i 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::realpart(value))); - else if (!x.imag().is_rational()) + else if (!instanceof(imagpart(value), cln::cl_RA_ring)) prec = cln::float_format(cln::the(cln::imagpart(value))); - // [Kol] (5.3) - if (cln::realpart(value) < -0.5) { + if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95))) { cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n) - * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p); + * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p); for (int s=0; s(x1) && is_a(x2) && is_a(x3)) { - return S_num(ex_to(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); + if (n.info(info_flags::posint) && p.info(info_flags::posint)) { + const int n_ = ex_to(n).to_int(); + const int p_ = ex_to(p).to_int(); + if (is_a(x)) { + const cln::cl_N x_ = ex_to(x).to_cl_N(); + const cln::cl_N result = S_num(n_, p_, x_); + return numeric(result); + } else { + ex x_val = x.evalf(); + if (is_a(x_val)) { + const cln::cl_N x_val_ = ex_to(x_val).to_cl_N(); + const cln::cl_N result = S_num(n_, p_, x_val_); + return numeric(result); + } + } } - return S(x1,x2,x3).hold(); + return S(n, p, x).hold(); } -static ex S_eval(const ex& x1, const ex& x2, const ex& x3) +static ex S_eval(const ex& n, const ex& p, const ex& x) { - if (x2 == 1) { - return Li(x1+1,x3); + if (n.info(info_flags::posint) && p.info(info_flags::posint)) { + if (x == 0) { + return _ex0; + } + if (x == 1) { + lst m(n+1); + for (int i=ex_to(p).to_int()-1; i>0; i--) { + m.append(1); + } + return zeta(m); + } + if (p == 1) { + return Li(n+1, x); + } + if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) { + int n_ = ex_to(n).to_int(); + int p_ = ex_to(p).to_int(); + const cln::cl_N x_ = ex_to(x).to_cl_N(); + const cln::cl_N result = S_num(n_, p_, x_); + return numeric(result); + } } - if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) && - x1.info(info_flags::posint) && x2.info(info_flags::posint)) { - return S_num(ex_to(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); + if (n.is_zero()) { + // [Kol] (5.3) + return pow(-log(1-x), p) / factorial(p); } - return S(x1,x2,x3).hold(); + return S(n, p, x).hold(); } -static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options) +static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options) { - epvector seq; - seq.push_back(expair(S(x1,x2,x3), 0)); - return pseries(rel,seq); + if (p == _ex1) { + return Li(n+1, x).series(rel, order, options); + } + + const ex x_pt = x.subs(rel, subs_options::no_pattern); + if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) { + // First special case: x==0 (derivatives have poles) + if (x_pt.is_zero()) { + const symbol s; + ex ser; + // manually construct the primitive expansion + // subsum = Euler-Zagier-Sum is needed + // dirty hack (slow ...) calculation of subsum: + std::vector presubsum, subsum; + subsum.push_back(0); + for (int i=1; i=1 (branch cut) + throw std::runtime_error("S_series: don't know how to do the series expansion at this point!"); + } + // all other cases should be safe, by now: + throw do_taylor(); // caught by function::series() } -static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param) +static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param) { GINAC_ASSERT(deriv_param < 3); if (deriv_param < 2) { return _ex0; } - if (x1 > 0) { - return S(x1-1, x2, x3) / x3; + if (n > 0) { + return S(n-1, p, x) / x; } else { - return S(x1, x2-1, x3) / (1-x3); + return S(n, p-1, x) / (1-x); } } @@ -979,12 +2211,12 @@ static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_con REGISTER_FUNCTION(S, - evalf_func(S_evalf). - eval_func(S_eval). - series_func(S_series). - derivative_func(S_deriv). - print_func(S_print_latex). - do_not_evalf_params()); + evalf_func(S_evalf). + eval_func(S_eval). + series_func(S_series). + derivative_func(S_deriv). + print_func(S_print_latex). + do_not_evalf_params()); ////////////////////////////////////////////////////////////////////// @@ -999,6 +2231,10 @@ REGISTER_FUNCTION(S, // anonymous namespace for helper functions namespace { + +// regulates the pole (used by 1/x-transformation) +symbol H_polesign("IMSIGN"); + // convert parameters from H to Li representation // parameters are expected to be in expanded form, i.e. only 0, 1 and -1 @@ -1045,7 +2281,7 @@ bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf) } } if (has_negative_parameters) { - for (int i=0; i 1; acc--) { - throw std::runtime_error("ERROR!"); - m.append(0); - } return has_negative_parameters; } @@ -1089,7 +2321,7 @@ struct map_trafo_H_convert_to_Li : public map_function s.let_op(0) = s.op(0) * arg; return pf * Li(m, s).hold(); } else { - for (int i=0; i(e.op(0))) { - parameter = ex_to(e.op(0)); + parameter = ex_to(e.op(0)); } else { parameter = lst(e.op(0)); } @@ -1171,7 +2403,7 @@ struct map_trafo_H_reduce_trailing_zeros : public map_function // parameter.remove_last(); - int lastentry = parameter.nops(); + std::size_t lastentry = parameter.nops(); while ((lastentry > 0) && (parameter[lastentry-1] == 0)) { lastentry--; } @@ -1210,32 +2442,32 @@ struct map_trafo_H_reduce_trailing_zeros : public map_function // returns an expression with zeta functions corresponding to the parameter list for H -ex convert_H_to_zeta(const lst& l) +ex convert_H_to_zeta(const lst& m) { symbol xtemp("xtemp"); map_trafo_H_reduce_trailing_zeros filter; map_trafo_H_convert_to_zeta filter2; - return filter2(filter(H(l, xtemp).hold())).subs(xtemp == 1); + return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1); } // convert signs form Li to H representation -// not used yet! -lst convert_parameter_Li_to_H(const lst& l, ex& pf) +lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf) { lst res; - lst::const_iterator it = l.begin(); - ex signum = *it; - pf = *it; - res.append(*it); - it++; - while (it != l.end()) { - signum = *it * signum; - res.append(signum); + lst::const_iterator itm = m.begin(); + lst::const_iterator itx = ++x.begin(); + int signum = 1; + pf = _ex1; + res.append(*itm); + itm++; + while (itx != x.end()) { + signum *= (*itx > 0) ? 1 : -1; pf *= signum; - it++; + res.append((*itm) * signum); + itm++; + itx++; } - return res; } @@ -1260,9 +2492,9 @@ ex trafo_H_mult(const ex& h1, const ex& h2) hlong = h2.op(0).op(0); } } - for (int i=0; i<=hlong.nops(); i++) { + for (std::size_t i=0; i<=hlong.nops(); i++) { lst newparameter; - int j=0; + std::size_t j=0; for (; j(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") { @@ -1324,7 +2556,7 @@ struct map_trafo_H_mult : public map_function if (Hlst.nops() > 0) { ex buffer = trafo_H_mult(firstH, Hlst.op(0)); result *= buffer; - for (int i=1; i(e.op(i))) { std::string name = ex_to(e.op(i)).get_name(); if (name == "H") { @@ -1372,6 +2604,37 @@ ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg) } +// do integration [ReV] (49) +// put parameter 1 in front of existing parameters +ex trafo_H_prepend_one(const ex& e, const ex& arg) +{ + ex h; + std::string name; + if (is_a(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (std::size_t i=0; i(e.op(i))) { + std::string name = ex_to(e.op(i)).get_name(); + if (name == "H") { + h = e.op(i); + } + } + } + } + if (h != 0) { + lst newparameter = ex_to(h.op(0)); + newparameter.prepend(1); + return e.subs(h == H(newparameter, h.op(1)).hold()); + } else { + return e * H(lst(1),1-arg).hold(); + } +} + + // do integration [ReV] (55) // put parameter -1 in front of existing parameters ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg) @@ -1384,7 +2647,7 @@ ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg) 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") { @@ -1417,7 +2680,7 @@ ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg) 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") { @@ -1448,7 +2711,7 @@ ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg) 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") { @@ -1467,6 +2730,107 @@ ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg) } +// 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); + + // special cases if all parameters are either 0, 1 or -1 + bool allthesame = true; + if (parameter.op(0) == 0) { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != 0) { + allthesame = false; + break; + } + } + if (allthesame) { + lst newparameter; + for (int i=parameter.nops(); i>0; i--) { + newparameter.append(1); + } + return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold(); + } + } else if (parameter.op(0) == -1) { + throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!"); + } else { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != 1) { + allthesame = false; + break; + } + } + if (allthesame) { + lst newparameter; + for (int i=parameter.nops(); i>0; i--) { + newparameter.append(0); + } + return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold(); + } + } + + lst newparameter = parameter; + newparameter.remove_first(); + + if (parameter.op(0) == 0) { + + // leading zero + ex res = convert_H_to_zeta(parameter); + //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1))); + map_trafo_H_1mx recursion; + ex buffer = recursion(H(newparameter, arg).hold()); + if (is_a(buffer)) { + for (std::size_t i = 0; i < buffer.nops(); i++) { + res -= trafo_H_prepend_one(buffer.op(i), arg); + } + } else { + res -= trafo_H_prepend_one(buffer, arg); + } + return res; + + } else { + + // leading one + map_trafo_H_1mx recursion; + map_trafo_H_mult unify; + ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold(); + std::size_t firstzero = 0; + while (parameter.op(firstzero) == 1) { + firstzero++; + } + for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) { + lst newparameter; + std::size_t j=0; + for (; j<=i; j++) { + newparameter.append(parameter[j+1]); + } + newparameter.append(1); + for (; j 1/x transformation struct map_trafo_H_1overx : public map_function { @@ -1486,7 +2850,7 @@ struct map_trafo_H_1overx : public map_function // 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(buffer)) { - for (int i=0; i(buffer)) { - for (int i=0; i& m, const cln::cl_N& x) static ex H_evalf(const ex& x1, const ex& x2) { - if (is_a(x1) && is_a(x2)) { - for (int i=0; i(x1)) { + + cln::cl_N x; + if (is_a(x2)) { + x = ex_to(x2).to_cl_N(); + } else { + ex x2_val = x2.evalf(); + if (is_a(x2_val)) { + x = ex_to(x2_val).to_cl_N(); + } + } + + for (std::size_t i = 0; i < x1.nops(); i++) { if (!x1.op(i).info(info_flags::integer)) { - return H(x1,x2).hold(); + return H(x1, x2).hold(); } } if (x1.nops() < 1) { - return H(x1,x2).hold(); + return H(x1, x2).hold(); } - cln::cl_N x = ex_to(x2).to_cl_N(); - const lst& morg = ex_to(x1); // remove trailing zeros ... if (*(--morg.end()) == 0) { @@ -1771,6 +3144,7 @@ static ex H_evalf(const ex& x1, const ex& x2) return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf(); } // ... and expand parameter notation + bool has_minus_one = false; lst m; for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) { if (*it > 1) { @@ -1778,19 +3152,18 @@ static ex H_evalf(const ex& x1, const ex& x2) m.append(0); } m.append(1); - } else if (*it < -1) { + } else if (*it <= -1) { for (ex count=*it+1; count < 0; count++) { m.append(0); } m.append(-1); + has_minus_one = true; } else { m.append(*it); } } - // since the transformations produce a lot of terms, they are only efficient for - // argument near one. - // no transformation needed -> do summation + // do summation if (cln::abs(x) < 0.95) { lst m_lst; lst s_lst; @@ -1820,12 +3193,13 @@ static ex H_evalf(const ex& x1, const ex& x2) } } + symbol xtemp("xtemp"); 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 + if (cln::abs(x) >= 2.0) { + 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 res.subs(xtemp == numeric(x)).evalf(); + } + + // check transformations for 0.95 <= |x| < 2.0 + + // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47 + if (cln::abs(x-9.53) <= 9.47) { // x -> (1-x)/(1+x) map_trafo_H_1mxt1px trafo; res *= trafo(H(m, xtemp)); } else { - // x -> 1/x - map_trafo_H_1overx trafo; + // x -> 1-x + if (has_minus_one) { + map_trafo_H_convert_to_Li filter; + return filter(H(m, numeric(x)).hold()).evalf(); + } + map_trafo_H_1mx trafo; res *= trafo(H(m, xtemp)); } - // 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(); } @@ -1862,54 +3243,157 @@ static ex H_evalf(const ex& x1, const ex& x2) } -static ex H_eval(const ex& x1, const ex& x2) +static ex H_eval(const ex& m_, const ex& x) { - if (x2 == 0) { - return 0; + lst m; + if (is_a(m_)) { + m = ex_to(m_); + } else { + m = lst(m_); } -//TODO -// if (x2 == 1) { -// return zeta(x1); -// } -// if (x1.nops() == 1) { -// return Li(x1.op(0), x2); -// } - if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) { - return H(x1,x2).evalf(); + if (m.nops() == 0) { + return _ex1; + } + ex pos1; + ex pos2; + ex n; + ex p; + int step = 0; + if (*m.begin() > _ex1) { + step++; + pos1 = _ex0; + pos2 = _ex1; + n = *m.begin()-1; + p = _ex1; + } else if (*m.begin() < _ex_1) { + step++; + pos1 = _ex0; + pos2 = _ex_1; + n = -*m.begin()-1; + p = _ex1; + } else if (*m.begin() == _ex0) { + pos1 = _ex0; + n = _ex1; + } else { + pos1 = *m.begin(); + p = _ex1; + } + for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) { + if ((*it).info(info_flags::integer)) { + if (step == 0) { + if (*it > _ex1) { + if (pos1 == _ex0) { + step = 1; + pos2 = _ex1; + n += *it-1; + p = _ex1; + } else { + step = 2; + } + } else if (*it < _ex_1) { + if (pos1 == _ex0) { + step = 1; + pos2 = _ex_1; + n += -*it-1; + p = _ex1; + } else { + step = 2; + } + } else { + if (*it != pos1) { + step = 1; + pos2 = *it; + } + if (*it == _ex0) { + n++; + } else { + p++; + } + } + } else if (step == 1) { + if (*it != pos2) { + step = 2; + } else { + if (*it == _ex0) { + n++; + } else { + p++; + } + } + } + } else { + // if some m_i is not an integer + return H(m_, x).hold(); + } } - return H(x1,x2).hold(); + if ((x == _ex1) && (*(--m.end()) != _ex0)) { + return convert_H_to_zeta(m); + } + if (step == 0) { + if (pos1 == _ex0) { + // all zero + if (x == _ex0) { + return H(m_, x).hold(); + } + return pow(log(x), m.nops()) / factorial(m.nops()); + } else { + // all (minus) one + return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops()); + } + } else if ((step == 1) && (pos1 == _ex0)){ + // convertible to S + if (pos2 == _ex1) { + return S(n, p, x); + } else { + return pow(-1, p) * S(n, p, -x); + } + } + if (x == _ex0) { + return _ex0; + } + if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) { + return H(m_, x).evalf(); + } + return H(m_, x).hold(); } -static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options) +static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options) { epvector seq; - seq.push_back(expair(H(x1,x2), 0)); - return pseries(rel,seq); + seq.push_back(expair(H(m, x), 0)); + return pseries(rel, seq); } -static ex H_deriv(const ex& x1, const ex& x2, unsigned deriv_param) +static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param) { GINAC_ASSERT(deriv_param < 2); if (deriv_param == 0) { return _ex0; } - if (is_a(x1)) { - lst newparameter = ex_to(x1); - if (x1.op(0) == 1) { - newparameter.remove_first(); - return 1/(1-x2) * H(newparameter, x2); - } else { - newparameter[0]--; - return H(newparameter, x2).hold() / x2; - } + lst m; + if (is_a(m_)) { + m = ex_to(m_); } else { - if (x1 == 1) { - return 1/(1-x2); - } else { - return H(x1-1, x2).hold() / x2; - } + m = lst(m_); + } + ex mb = *m.begin(); + if (mb > _ex1) { + m[0]--; + return H(m, x) / x; + } + if (mb < _ex_1) { + m[0]++; + return H(m, x) / x; + } + m.remove_first(); + if (mb == _ex1) { + return 1/(1-x) * H(m, x); + } else if (mb == _ex_1) { + return 1/(1+x) * H(m, x); + } else { + return H(m, x) / x; } } @@ -1937,23 +3421,23 @@ static void H_print_latex(const ex& m_, const ex& x, const print_context& c) REGISTER_FUNCTION(H, - evalf_func(H_evalf). - eval_func(H_eval). - series_func(H_series). - derivative_func(H_deriv). - print_func(H_print_latex). - do_not_evalf_params()); + evalf_func(H_evalf). + eval_func(H_eval). + series_func(H_series). + derivative_func(H_deriv). + print_func(H_print_latex). + do_not_evalf_params()); // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms -ex convert_H_to_Li(const ex& parameterlst, const ex& arg) +ex convert_H_to_Li(const ex& m, const ex& x) { map_trafo_H_reduce_trailing_zeros filter; map_trafo_H_convert_to_Li filter2; - if (is_a(parameterlst)) { - return filter2(filter(H(parameterlst, arg).hold())).eval(); + if (is_a(m)) { + return filter2(filter(H(m, x).hold())); } else { - return filter2(filter(H(lst(parameterlst), arg).hold())).eval(); + return filter2(filter(H(lst(m), x).hold())); } } @@ -1973,13 +3457,6 @@ 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) { @@ -1994,44 +3471,39 @@ void halfcyclic_convolute(const std::vector& a, const std::vector& s) +static void initcX(std::vector& crX, + const std::vector& s, + const int L2) { - 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)); - } + std::vector crB(L2 + 1); + for (int i=0; i<=L2; i++) + crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i); int Sm = 0; int Smp1 = 0; - for (int m=0; m crGbuf; - Sm = Sm + s[m]; + std::vector > crG(s.size() - 1, std::vector(L2 + 1)); + for (int m=0; m < s.size() - 1; m++) { + 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); + for (int i = 0; i <= L2; i++) + crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2); } crX = crB; - for (int m=0; m Xbuf; - for (int i=0; i<=L2; i++) { - Xbuf.push_back(crX[i] * crG[m][i]); - } + for (std::size_t m = 0; m < s.size() - 1; m++) { + std::vector Xbuf(L2 + 1); + for (int i = 0; i <= L2; i++) + Xbuf[i] = crX[i] * crG[m][i]; + halfcyclic_convolute(Xbuf, crB, crX); } } // [Cra] section 4 -cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk) +static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk, + const std::vector& crX) { cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); cln::cl_N factor = cln::expt(lambda, Sqk); @@ -2049,11 +3521,9 @@ cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk) // [Cra] section 4 -void calc_f(int maxr) +static void calc_f(std::vector >& f_kj, + const int maxr, const int L1) { - 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(); @@ -2079,7 +3549,8 @@ void calc_f(int maxr) // [Cra] (3.1) -cln::cl_N crandall_Z(const std::vector& s) +static cln::cl_N crandall_Z(const std::vector& s, + const std::vector >& f_kj) { const int j = s.size(); @@ -2120,6 +3591,8 @@ cln::cl_N zeta_do_sum_Crandall(const std::vector& s) std::vector r = s; const int j = r.size(); + std::size_t L1; + // decide on maximal size of f_kj for crandall_Z if (Digits < 50) { L1 = 150; @@ -2127,6 +3600,7 @@ cln::cl_N zeta_do_sum_Crandall(const std::vector& s) L1 = Digits * 3 + j*2; } + std::size_t L2; // decide on maximal size of crX for crandall_Y if (Digits < 38) { L2 = 63; @@ -2153,7 +3627,8 @@ cln::cl_N zeta_do_sum_Crandall(const std::vector& s) } } - calc_f(maxr); + std::vector > f_kj(L1); + calc_f(f_kj, maxr, L1); const cln::cl_N r0factorial = cln::factorial(r[0]-1); @@ -2167,12 +3642,13 @@ cln::cl_N zeta_do_sum_Crandall(const std::vector& s) Srun -= skp1buf; r.pop_back(); - initcX(r); + std::vector crX; + initcX(crX, r, L2); for (int q=0; q& s) } rz.insert(rz.begin(), r.back()); - initcX(rz); + std::vector crX; + initcX(crX, rz, L2); - res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz); + res = (res + crandall_Y_loop(S-j, crX)) / r0factorial + + crandall_Z(rz, f_kj); return res; } @@ -2230,7 +3708,7 @@ cln::cl_N zeta_do_Hoelder_convolution(const std::vector& m_, const std::vec s_p[0] = s_p[0] * cln::cl_N("1/2"); // convert notations int sig = 1; - for (int i=0; i& m_, const std::vec if (m_p.size() == 0) break; res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q); - + } while (true); // last term res = res + signum * multipleLi_do_sum(m_q, s_q); - + return res; } @@ -2350,7 +3828,7 @@ static ex zeta1_evalf(const ex& x) return numeric(zeta_do_sum_simple(r)); } } - + // single zeta value if (is_exactly_a(x) && (x != 1)) { try { @@ -2362,87 +3840,86 @@ 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(); + if (y.is_equal(*_num1_p)) { + 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); + return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y); } } else { if (y.info(info_flags::odd)) { - return -bernoulli(_num1-y) / (_num1-y); + return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y); } else { return _ex0; } } } // zeta(float) - if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) - return zeta1_evalf(x); + if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) { + return zeta1_evalf(m); + } } - return zeta(x).hold(); + return zeta(m).hold(); } -static ex zeta1_deriv(const ex& x, unsigned deriv_param) +static ex zeta1_deriv(const ex& m, unsigned deriv_param) { GINAC_ASSERT(deriv_param==0); - if (is_exactly_a(x)) { + if (is_exactly_a(m)) { return _ex0; } else { - return zeta(_ex1, x); + return zetaderiv(_ex1, m); } } -static void zeta1_print_latex(const ex& x, const print_context& c) +static void zeta1_print_latex(const ex& m_, const print_context& c) { c.s << "\\zeta("; - if (is_a(x)) { - lst arg; - arg = ex_to(x); - lst::const_iterator it = arg.begin(); + if (is_a(m_)) { + const lst& m = ex_to(m_); + lst::const_iterator it = m.begin(); (*it).print(c); it++; - for (; it != arg.end(); it++) { + for (; it != m.end(); it++) { c.s << ","; (*it).print(c); } } else { - x.print(c); + m_.print(c); } c.s << ")"; } -unsigned zeta1_SERIAL::serial = - function::register_new(function_options("zeta"). - evalf_func(zeta1_evalf). - eval_func(zeta1_eval). - derivative_func(zeta1_deriv). - print_func(zeta1_print_latex). - do_not_evalf_params(). - overloaded(2)); +unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1). + evalf_func(zeta1_evalf). + eval_func(zeta1_eval). + derivative_func(zeta1_deriv). + print_func(zeta1_print_latex). + do_not_evalf_params(). + overloaded(2)); ////////////////////////////////////////////////////////////////////// @@ -2494,96 +3971,92 @@ static ex zeta2_evalf(const ex& x, const ex& s) // use Hoelder convolution return numeric(zeta_do_Hoelder_convolution(xi, si)); } - + return zeta(x, s).hold(); } -static ex zeta2_eval(const ex& x, const ex& s) +static ex zeta2_eval(const ex& m, const ex& s_) { - if (is_exactly_a(s)) { - const lst& l = ex_to(s); - lst::const_iterator it = l.begin(); - while (it != l.end()) { - if ((*it).info(info_flags::negative)) { - return zeta(x, s).hold(); + if (is_exactly_a(s_)) { + const lst& s = ex_to(s_); + for (lst::const_iterator it = s.begin(); it != s.end(); it++) { + if ((*it).info(info_flags::positive)) { + continue; } - it++; - } - return zeta(x); - } else { - if (s.info(info_flags::positive)) { - return zeta(x); + return zeta(m, s_).hold(); } + return zeta(m); + } else if (s_.info(info_flags::positive)) { + return zeta(m); } - return zeta(x, s).hold(); + return zeta(m, s_).hold(); } -static ex zeta2_deriv(const ex& x, const ex& s, unsigned deriv_param) +static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param) { GINAC_ASSERT(deriv_param==0); - if (is_exactly_a(x)) { + if (is_exactly_a(m)) { return _ex0; } else { - if ((is_exactly_a(s) && (s.op(0) > 0)) || (s > 0)) { - return zeta(_ex1, x); + if ((is_exactly_a(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) { + return zetaderiv(_ex1, m); } return _ex0; } } -static void zeta2_print_latex(const ex& x, const ex& s, const print_context& c) +static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c) { - lst arg; - if (is_a(x)) { - arg = ex_to(x); + lst m; + if (is_a(m_)) { + m = ex_to(m_); } else { - arg = lst(x); + m = lst(m_); } - lst sig; - if (is_a(s)) { - sig = ex_to(s); + lst s; + if (is_a(s_)) { + s = ex_to(s_); } else { - sig = lst(s); + s = lst(s_); } c.s << "\\zeta("; - lst::const_iterator itarg = arg.begin(); - lst::const_iterator itsig = sig.begin(); - if (*itsig < 0) { + lst::const_iterator itm = m.begin(); + lst::const_iterator its = s.begin(); + if (*its < 0) { c.s << "\\overline{"; - (*itarg).print(c); + (*itm).print(c); c.s << "}"; } else { - (*itarg).print(c); + (*itm).print(c); } - itsig++; - itarg++; - for (; itarg != arg.end(); itarg++, itsig++) { + its++; + itm++; + for (; itm != m.end(); itm++, its++) { c.s << ","; - if (*itsig < 0) { + if (*its < 0) { c.s << "\\overline{"; - (*itarg).print(c); + (*itm).print(c); c.s << "}"; } else { - (*itarg).print(c); + (*itm).print(c); } } c.s << ")"; } -unsigned zeta2_SERIAL::serial = - function::register_new(function_options("zeta"). - evalf_func(zeta2_evalf). - eval_func(zeta2_eval). - derivative_func(zeta2_deriv). - print_func(zeta2_print_latex). - do_not_evalf_params(). - overloaded(2)); +unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2). + evalf_func(zeta2_evalf). + eval_func(zeta2_eval). + derivative_func(zeta2_deriv). + print_func(zeta2_print_latex). + do_not_evalf_params(). + overloaded(2)); } // namespace GiNaC