X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_nstdsums.cpp;h=c20526bafd5a38fb301d9470e8c4138ba8540d27;hp=17db2c885dfd63a8ea880d61ff85ec4be5e8ea0f;hb=0eaae44cd9eb9fa987bb9cbd4341b0f4c8d2f495;hpb=fab89324be435eb99b50b9ee31ea4408843ab704 diff --git a/ginac/inifcns_nstdsums.cpp b/ginac/inifcns_nstdsums.cpp index 17db2c88..c20526ba 100644 --- a/ginac/inifcns_nstdsums.cpp +++ b/ginac/inifcns_nstdsums.cpp @@ -4,11 +4,12 @@ * * 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) - * 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)) + * multiple polylogarithm Li(lst{m_1,...,m_k},lst{x_1,...,x_k}) + * G(lst{a_1,...,a_k},y) or G(lst{a_1,...,a_k},lst{s_1,...,s_k},y) + * Nielsen's generalized polylogarithm S(n,p,x) + * harmonic polylogarithm H(m,x) or H(lst{m_1,...,m_k},x) + * multiple zeta value zeta(m) or zeta(lst{m_1,...,m_k}) + * alternating Euler sum zeta(m,s) or zeta(lst{m_1,...,m_k},lst{s_1,...,s_k}) * * Some remarks: * @@ -17,36 +18,36 @@ * [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. * - * - 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. + * - All functions can be numerically evaluated with arguments in the whole complex plane. The parameters + * for Li, zeta and S must be positive integers. If you want to have an alternating Euler sum, you have + * to give the signs of the parameters as a second argument s to zeta(m,s) containing 1 and -1. * * - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and * look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in - * [Cra] and [BBB] for speed up. + * [Cra] and [BBB] for speed up. Multiple polylogarithms use Hoelder convolution [BBB]. * - * - 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. + * - 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) 1999-2004 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2019 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 @@ -60,13 +61,9 @@ * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software - * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ -#include -#include -#include - #include "inifcns.h" #include "add.h" @@ -82,6 +79,11 @@ #include "utils.h" #include "wildcard.h" +#include +#include +#include +#include +#include namespace GiNaC { @@ -101,7 +103,7 @@ namespace { // lookup table for factors built from Bernoulli numbers // see fill_Xn() -std::vector > Xn; +std::vector> Xn; // initial size of Xn that should suffice for 32bit machines (must be even) const int xninitsizestep = 26; int xninitsize = xninitsizestep; @@ -123,7 +125,7 @@ void fill_Xn(int n) if (n>1) { // calculate X_2 and higher (corresponding to Li_4 and higher) std::vector buf(xninitsize); - std::vector::iterator it = buf.begin(); + auto 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++; @@ -148,7 +150,7 @@ void fill_Xn(int n) } else if (n==1) { // special case to handle the X_0 correct std::vector buf(xninitsize); - std::vector::iterator it = buf.begin(); + auto it = buf.begin(); cln::cl_N result; *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1 it++; @@ -172,7 +174,7 @@ void fill_Xn(int n) } else { // calculate X_0 std::vector buf(xninitsize/2); - std::vector::iterator it = buf.begin(); + auto it = buf.begin(); for (int i=1; i<=xninitsize/2; i++) { *it = bernoulli(i*2).to_cl_N(); it++; @@ -183,7 +185,6 @@ void fill_Xn(int n) xnsize++; } - // doubles the number of entries in each Xn[] void double_Xn() { @@ -192,7 +193,7 @@ void double_Xn() for (int i=1; i<=xninitsizestep/2; ++i) { Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N()); } - if (Xn.size() > 0) { + if (Xn.size() > 1) { int xend = xninitsize + xninitsizestep; cln::cl_N result; // X_1 @@ -209,7 +210,7 @@ void double_Xn() } } // X_n - for (int n=2; n::const_iterator xend = Xn[0].end(); cln::cl_N u = -cln::log(1-x); cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits)); - cln::cl_N res = u - u*u/4; + 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; i++; if (++it == xend) { @@ -318,7 +320,7 @@ cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x) // 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 @@ -336,16 +338,20 @@ cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& pr // the switching point was empirically determined. the optimal point // depends on hardware, Digits, ... so an approx value is okay. // it solves also the problem with precision due to the u=-log(1-x) transformation - if (cln::abs(cln::realpart(x)) < 0.25) { - + if (cln::abs(x) < 0.25) { return Li2_do_sum(x); } else { + // Li2_do_sum practically doesn't converge near x == ±I return Li2_do_sum_Xn(x); } } else { // choose the faster algorithm if (cln::abs(cln::realpart(x)) > 0.75) { - return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2); + if ( x == 1 ) { + return cln::zeta(2); + } else { + return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2); + } } else { return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2); } @@ -361,15 +367,17 @@ cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& pr if (cln::realpart(x) < 0.5) { // choose the faster algorithm // with n>=12 the "normal" summation always wins against the method with Xn - if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) { + if ((cln::abs(x) < 0.3) || (n >= 12)) { return Lin_do_sum(n, x); } else { + // Li2_do_sum practically doesn't converge near x == ±I return Lin_do_sum_Xn(n, x); } } else { - cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1); + cln::cl_N result = 0; + if ( x != 1 ) result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1); for (int j=0; j(cln::realpart(value))); - else if (!x.imag().is_rational()) + else if (!instanceof(imagpart(x), cln::cl_RA_ring)) prec = cln::float_format(cln::the(cln::imagpart(value))); // [Kol] (5.15) @@ -430,7 +445,7 @@ 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 (const auto & it : x) { + 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)); @@ -468,159 +490,1217 @@ 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--) { - t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]); + q++; + t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one; + for (int k=j-2; k>=0; k--) { + flag_accidental_zero = cln::zerop(t[k+1]); + t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]); + } + } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero ); + + return t[0]; +} + + +// 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 (const auto & it : a) { + 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(a.begin()+1, a.end()); + ex result = G_eval1(a.front(), scale, gsyms) * G_eval(short_a, scale, gsyms); + + auto it = short_a.begin(); + advance(it, count_ones-1); + for (; it != short_a.end(); ++it) { + + Gparameter newa(short_a.begin(), it); + newa.push_back(*it); + newa.push_back(a[0]); + newa.insert(newa.end(), it+1, short_a.end()); + result -= G_eval(newa, scale, gsyms); + } + return result / count_ones; + } + + // G({1,...,1};y) -> 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 (const auto & it : a) { + 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); +} + +// convert back to standard G-function, keep information on small imaginary parts +ex G_eval_to_G(const Gparameter& a, int scale, const exvector& gsyms) +{ + lst z; + lst s; + for (const auto & it : a) { + if (it != 0) { + z.append(gsyms[std::abs(it)]); + if ( it<0 ) { + s.append(-1); + } else { + s.append(1); + } + } else { + z.append(0); + s.append(1); + } + } + return G(z,s,gsyms[std::abs(scale)]); +} + + +// 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(); + auto lastnonzero = a.end(); + for (auto it = a.begin(); it != a.end(); ++it) { + if (std::abs(*it) > 0) { + ++depth; + trailing_zeros = 0; + lastnonzero = it; + if (std::abs(*it) < scale) { + convergent = false; + if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) { + min_it = it; + } + } + } else { + ++trailing_zeros; + } + } + if (lastnonzero == a.end()) + return a.end(); + return ++lastnonzero; +} + + +// 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 (auto 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, bool flag_trailing_zeros_only); + + +// G transformation [VSW] +ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale, + const exvector& gsyms, bool flag_trailing_zeros_only) +{ + // main recursion routine + // + // pendint = ( y1, b1, ..., br ) + // a = ( a1, ..., amin, ..., aw ) + // scale = y2 + // + // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2) + // where sr replaces amin + + // find smallest alpha, determine depth and trailing zeros, and check for convergence + bool convergent; + int depth, trailing_zeros; + Gparameter::const_iterator min_it; + auto firstzero = check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it); + int min_it_pos = distance(a.begin(), min_it); + + // special case: all a's are zero + if (depth == 0) { + ex result; + + if (a.size() == 0) { + result = 1; + } else { + result = G_eval(a, scale, gsyms); + } + if (pendint.size() > 0) { + result *= trailing_zeros_G(convert_pending_integrals_G(pendint), + pendint.front(), + gsyms); + } + return result; + } + + // handle trailing zeros + if (trailing_zeros > 0) { + ex result; + Gparameter new_a(a.begin(), a.end()-1); + result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only); + for (auto it = a.begin(); it != firstzero; ++it) { + Gparameter new_a(a.begin(), it); + new_a.push_back(0); + new_a.insert(new_a.end(), it, a.end()-1); + result -= G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only); + } + return result / trailing_zeros; + } + + // flag_trailing_zeros_only: in this case we don't have pending integrals + if (flag_trailing_zeros_only) + return G_eval_to_G(a, scale, gsyms); + + // 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, flag_trailing_zeros_only)* + G_transform(empty, a1, scale, gsyms, flag_trailing_zeros_only); + + result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms, flag_trailing_zeros_only); + return result; + } + + Gparameter empty; + Gparameter::iterator changeit; + + // first term G(a_1,..,0,...,a_w;a_0) + Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]); + Gparameter new_a = a; + new_a[min_it_pos] = 0; + ex result = G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only); + if (pendint.size() > 0) { + result *= trailing_zeros_G(convert_pending_integrals_G(pendint), + pendint.front(), gsyms); + } + + // other terms + changeit = new_a.begin() + min_it_pos; + changeit = new_a.erase(changeit); + if (changeit != new_a.begin()) { + // smallest in the middle + new_pendint.push_back(*changeit); + result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only); + int buffer = *changeit; + *changeit = *min_it; + result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only); + *changeit = buffer; + new_pendint.pop_back(); + --changeit; + new_pendint.push_back(*changeit); + result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only); + *changeit = *min_it; + result -= G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only); + } else { + // smallest at the front + new_pendint.push_back(scale); + result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only); + new_pendint.back() = *changeit; + result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only); + *changeit = *min_it; + result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only); + } + return result; +} + + +// shuffles the two parameter list a1 and a2 and calls G_transform for every term except +// for the one that is equal to a_old +ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2, + const Gparameter& pendint, const Gparameter& a_old, int scale, + const exvector& gsyms, bool flag_trailing_zeros_only) +{ + if (a1.size()==0 && a2.size()==0) { + // veto the one configuration we don't want + if ( a0 == a_old ) return 0; + + return G_transform(pendint, a0, scale, gsyms, flag_trailing_zeros_only); + } + + if (a2.size()==0) { + Gparameter empty; + Gparameter aa0 = a0; + aa0.insert(aa0.end(),a1.begin(),a1.end()); + return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only); + } + + if (a1.size()==0) { + Gparameter empty; + Gparameter aa0 = a0; + aa0.insert(aa0.end(),a2.begin(),a2.end()); + return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only); + } + + Gparameter a1_removed(a1.begin()+1,a1.end()); + Gparameter a2_removed(a2.begin()+1,a2.end()); + + Gparameter a01 = a0; + Gparameter a02 = a0; + + a01.push_back( a1[0] ); + a02.push_back( a2[0] ); + + return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms, flag_trailing_zeros_only) + + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms, flag_trailing_zeros_only); +} + +// handles the transformations and the numerical evaluation of G +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_numeric(const std::vector& x, const std::vector& s, + const cln::cl_N& y); + +// do acceleration transformation (hoelder convolution [BBB]) +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_do_hoelder(std::vector x, /* yes, it's passed by value */ + const std::vector& s, const cln::cl_N& y) +{ + cln::cl_N result; + const std::size_t size = x.size(); + for (std::size_t i = 0; i < size; ++i) + x[i] = x[i]/y; + + for (std::size_t r = 0; r <= size; ++r) { + cln::cl_N buffer(1 & r ? -1 : 1); + cln::cl_RA p(2); + bool adjustp; + do { + adjustp = false; + for (std::size_t i = 0; i < size; ++i) { + if (x[i] == cln::cl_RA(1)/p) { + p = p/2 + cln::cl_RA(3)/2; + adjustp = true; + continue; + } + } + } while (adjustp); + cln::cl_RA q = p/(p-1); + std::vector qlstx; + std::vector qlsts; + for (std::size_t j = r; j >= 1; --j) { + qlstx.push_back(cln::cl_N(1) - x[j-1]); + if (imagpart(x[j-1])==0 && realpart(x[j-1]) >= 1) { + qlsts.push_back(1); + } else { + qlsts.push_back(-s[j-1]); + } + } + if (qlstx.size() > 0) { + buffer = buffer*G_numeric(qlstx, qlsts, 1/q); + } + std::vector plstx; + std::vector plsts; + for (std::size_t j = r+1; j <= size; ++j) { + plstx.push_back(x[j-1]); + plsts.push_back(s[j-1]); + } + if (plstx.size() > 0) { + buffer = buffer*G_numeric(plstx, plsts, 1/p); + } + result = result + buffer; + } + return result; +} + +class less_object_for_cl_N +{ +public: + bool operator() (const cln::cl_N & a, const cln::cl_N & b) const + { + // absolute value? + if (abs(a) != abs(b)) + return (abs(a) < abs(b)) ? true : false; + + // complex phase? + if (phase(a) != phase(b)) + return (phase(a) < phase(b)) ? true : false; + + // equal, therefore "less" is not true + return false; + } +}; + + +// convergence transformation, used for numerical evaluation of G function. +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_do_trafo(const std::vector& x, const std::vector& s, + const cln::cl_N& y, bool flag_trailing_zeros_only) +{ + // sort (|x|<->position) to determine indices + typedef std::multimap sortmap_t; + sortmap_t sortmap; + std::size_t size = 0; + for (std::size_t i = 0; i < x.size(); ++i) { + if (!zerop(x[i])) { + sortmap.insert(std::make_pair(x[i], i)); + ++size; + } + } + // include upper limit (scale) + sortmap.insert(std::make_pair(y, x.size())); + + // generate missing dummy-symbols + int i = 1; + // holding dummy-symbols for the G/Li transformations + exvector gsyms; + gsyms.push_back(symbol("GSYMS_ERROR")); + cln::cl_N lastentry(0); + for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) { + if (it != sortmap.begin()) { + if (it->second < x.size()) { + if (x[it->second] == lastentry) { + gsyms.push_back(gsyms.back()); + continue; + } + } else { + if (y == lastentry) { + gsyms.push_back(gsyms.back()); + continue; + } + } + } + std::ostringstream os; + os << "a" << i; + gsyms.push_back(symbol(os.str())); + ++i; + if (it->second < x.size()) { + lastentry = x[it->second]; + } else { + lastentry = y; + } + } + + // fill position data according to sorted indices and prepare substitution list + Gparameter a(x.size()); + exmap subslst; + std::size_t pos = 1; + int scale = pos; + for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) { + if (it->second < x.size()) { + if (s[it->second] > 0) { + a[it->second] = pos; + } else { + a[it->second] = -int(pos); + } + subslst[gsyms[pos]] = numeric(x[it->second]); + } else { + scale = pos; + subslst[gsyms[pos]] = numeric(y); + } + ++pos; + } + + // do transformation + Gparameter pendint; + ex result = G_transform(pendint, a, scale, gsyms, flag_trailing_zeros_only); + // replace dummy symbols with their values + result = result.expand(); + result = result.subs(subslst).evalf(); + if (!is_a(result)) + throw std::logic_error("G_do_trafo: G_transform returned non-numeric result"); + + cln::cl_N ret = ex_to(result).to_cl_N(); + return ret; +} + +// handles the transformations and the numerical evaluation of G +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_numeric(const std::vector& x, const std::vector& s, + const cln::cl_N& y) +{ + // check for convergence and necessary accelerations + bool need_trafo = false; + bool need_hoelder = false; + bool have_trailing_zero = false; + std::size_t depth = 0; + for (auto & xi : x) { + if (!zerop(xi)) { + ++depth; + const cln::cl_N x_y = abs(xi) - 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(xi/y) - 1) < 0.01) + need_hoelder = true; + } + } + if (zerop(x.back())) { + have_trailing_zero = true; + need_trafo = true; + } + + if (depth == 1 && x.size() == 2 && !need_trafo) + return - Li_projection(2, y/x[1], cln::float_format(Digits)); + + // do acceleration transformation (hoelder convolution [BBB]) + if (need_hoelder && !have_trailing_zero) + return G_do_hoelder(x, s, y); + + // convergence transformation + if (need_trafo) + return G_do_trafo(x, s, y, have_trailing_zero); + + // do summation + std::vector newx; + newx.reserve(x.size()); + std::vector m; + m.reserve(x.size()); + int mcount = 1; + int sign = 1; + cln::cl_N factor = y; + for (auto & xi : x) { + if (zerop(xi)) { + ++mcount; + } else { + newx.push_back(factor/xi); + factor = xi; + 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 (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { + for (int i = 1; i < *itm; ++i) { + newx.push_back(cln::cl_N(0)); + s.push_back(1); + } + const cln::cl_N xi = ex_to(*itx).to_cl_N(); + factor = factor/xi; + newx.push_back(factor); + if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) { + s.push_back(-1); + } + else { + s.push_back(1); + } + } + return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1))); +} + + +} // end of anonymous namespace + + +////////////////////////////////////////////////////////////////////// +// +// Generalized multiple polylogarithm G(x, y) and G(x, s, y) +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// + + +static ex G2_evalf(const ex& x_, const ex& y) +{ + if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) { + return G(x_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst{x_}; + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, y).hold(); + } + std::vector s; + s.reserve(x.nops()); + bool all_zero = true; + for (const auto & it : x) { + 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 (const auto & it : x) + xv.push_back(ex_to(it).to_cl_N()); + cln::cl_N result = G_numeric(xv, s, ex_to(y).to_cl_N()); + return numeric(result); +} + + +static ex G2_eval(const ex& x_, const ex& y) +{ + //TODO eval to MZV or H or S or Lin + + if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) { + return G(x_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst{x_}; + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, y).hold(); + } + std::vector s; + s.reserve(x.nops()); + bool all_zero = true; + bool crational = true; + for (const auto & it : x) { + 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 (const auto & it : x) + xv.push_back(ex_to(it).to_cl_N()); + cln::cl_N result = G_numeric(xv, s, ex_to(y).to_cl_N()); + return numeric(result); +} + + +// option do_not_evalf_params() removed. +unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2). + evalf_func(G2_evalf). + eval_func(G2_eval). + overloaded(2)); +//TODO +// derivative_func(G2_deriv). +// print_func(G2_print_latex). + + +static ex G3_evalf(const ex& x_, const ex& s_, const ex& y) +{ + if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) { + return G(x_, s_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst{x_}; + lst s = is_a(s_) ? ex_to(s_) : lst{s_}; + if (x.nops() != s.nops()) { + return G(x_, s_, y).hold(); + } + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, s_, y).hold(); + } + std::vector sn; + sn.reserve(s.nops()); + bool all_zero = true; + for (auto itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) { + if (!(*itx).info(info_flags::numeric)) { + return G(x_, y).hold(); + } + if (!(*its).info(info_flags::real)) { + return G(x_, y).hold(); + } + if (*itx != _ex0) { + all_zero = false; + } + if ( ex_to(*itx).is_real() ) { + if ( ex_to(*itx).is_positive() ) { + if ( *its >= 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } else { + sn.push_back(1); + } + } + else { + if ( ex_to(*itx).imag() > 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + std::vector xn; + xn.reserve(x.nops()); + for (const auto & it : x) + xn.push_back(ex_to(it).to_cl_N()); + cln::cl_N result = G_numeric(xn, sn, ex_to(y).to_cl_N()); + return numeric(result); +} + + +static ex G3_eval(const ex& x_, const ex& s_, const ex& y) +{ + //TODO eval to MZV or H or S or Lin + + if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) { + return G(x_, s_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst{x_}; + lst s = is_a(s_) ? ex_to(s_) : lst{s_}; + if (x.nops() != s.nops()) { + return G(x_, s_, y).hold(); + } + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, s_, y).hold(); + } + std::vector sn; + sn.reserve(s.nops()); + bool all_zero = true; + bool crational = true; + for (auto 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(); } - } while (t[0] != t0buf); - - return t[0]; + if (!(*itx).info(info_flags::crational)) { + crational = false; + } + if (*itx != _ex0) { + all_zero = false; + } + if ( ex_to(*itx).is_real() ) { + if ( ex_to(*itx).is_positive() ) { + if ( *its >= 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } else { + sn.push_back(1); + } + } + else { + if ( ex_to(*itx).imag() > 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + if (!y.info(info_flags::crational)) { + crational = false; + } + if (crational) { + return G(x_, s_, y).hold(); + } + std::vector xn; + xn.reserve(x.nops()); + for (const auto & it : x) + 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); } -// forward declaration for Li_eval() -lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf); - -} // end of anonymous namespace +// option do_not_evalf_params() removed. +// This is safe: in the code above it only matters if s_ > 0 or s_ < 0, +// s_ is allowed to be of floating type. +unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3). + evalf_func(G3_evalf). + eval_func(G3_eval). + overloaded(2)); +//TODO +// derivative_func(G3_deriv). +// print_func(G3_print_latex). ////////////////////////////////////////////////////////////////////// // -// Classical polylogarithm and multiple polylogarithm Li(n,x) +// Classical polylogarithm and multiple polylogarithm Li(m,x) // // GiNaC function // ////////////////////////////////////////////////////////////////////// -static ex Li_evalf(const ex& x1, const ex& x2) +static ex Li_evalf(const ex& m_, const ex& x_) { // classical polylogs - if (is_a(x1) && is_a(x2)) { - return Li_num(ex_to(x1).to_int(), ex_to(x2)); + 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 - else if (is_a(x1) && is_a(x2)) { - ex conv = 1; - for (int i=0; i(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 (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { + if (!(*itm).info(info_flags::posint)) { + return Li(m_, x_).hold(); } - if (!is_a(x2.op(i))) { - return Li(x1, x2).hold(); + if (!(*itx).info(info_flags::numeric)) { + return Li(m_, x_).hold(); } - conv *= x2.op(i); - if (abs(conv) >= 1) { - return Li(x1, x2).hold(); + if (*itx == _ex0) { + return _ex0; } } - std::vector m; - std::vector x; - for (int i=0; i(x1.nops()).to_int(); i++) { - m.push_back(ex_to(x1.op(i)).to_int()); - x.push_back(ex_to(x2.op(i)).to_cl_N()); - } - - return numeric(multipleLi_do_sum(m, x)); + return mLi_numeric(m, x); } - return Li(x1,x2).hold(); + return Li(m_,x_).hold(); } static ex Li_eval(const ex& m_, const ex& x_) { - if (m_.nops() < 2) { - ex m; - if (is_a(m_)) { - m = m_.op(0); - } else { - m = m_; - } - ex x; + if (is_a(m_)) { if (is_a(x_)) { - x = x_.op(0); - } else { - x = x_; - } - 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.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) { - return Li_num(ex_to(m).to_int(), ex_to(x)); - } - } else { - bool ish = true; - bool iszeta = true; - bool iszero = false; - bool doevalf = false; - bool doevalfveto = true; - const lst& m = ex_to(m_); - const lst& x = ex_to(x_); - lst::const_iterator itm = m.begin(); - lst::const_iterator itx = x.begin(); - for (; itm != m.end(); itm++, itx++) { - if (!(*itm).info(info_flags::posint)) { - return Li(m_, x_).hold(); + // 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; } - if ((*itx != _ex1) && (*itx != _ex_1)) { - if (itx != x.begin()) { - ish = false; + bool is_H = true; + bool is_zeta = true; + bool do_evalf = true; + bool crational = true; + for (auto 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; } - iszeta = false; } - if (*itx == _ex0) { - iszero = true; + if (is_zeta) { + lst newx; + for (const auto & itx : x) { + GINAC_ASSERT((itx == _ex1) || (itx == _ex_1)); + // XXX: 1 + 0.0*I is considered equal to 1. However + // the former is a not automatically converted + // to a real number. Do the conversion explicitly + // to avoid the "numeric::operator>(): complex inequality" + // exception (and similar problems). + newx.append(itx != _ex_1 ? _ex1 : _ex_1); + } + return zeta(m_, newx); } - if (!(*itx).info(info_flags::numeric)) { - doevalfveto = false; + if (is_H) { + ex prefactor; + lst newm = convert_parameter_Li_to_H(m, x, prefactor); + return prefactor * H(newm, x[0]); } - if (!(*itx).info(info_flags::crational)) { - doevalf = true; + if (do_evalf && !crational) { + return mLi_numeric(m,x); } } - if (iszeta) { - return zeta(m_, x_); - } - if (iszero) { - return _ex0; - } - if (ish) { - ex pf; - lst newm = convert_parameter_Li_to_H(m, x, pf); - return pf * H(newm, x[0]); + 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 (doevalfveto && doevalf) { - return Li(m_, x_).evalf(); + if (x_.is_equal(-I)) { + return power(Pi,_ex2)/_ex_48 - Catalan*I; } } + if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) { + int m__ = ex_to(m_).to_int(); + const cln::cl_N x__ = ex_to(x_).to_cl_N(); + const cln::cl_N result = Lin_numeric(m__, x__); + return numeric(result); + } + return Li(m_, x_).hold(); } static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options) { - epvector seq; - seq.push_back(expair(Li(m, x), 0)); - return pseries(rel, seq); + if (is_a(m) || is_a(x)) { + // multiple polylog + epvector seq { expair(Li(m, x), 0) }; + return pseries(rel, std::move(seq)); + } + + // classical polylog + const ex x_pt = x.subs(rel, subs_options::no_pattern); + if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) { + // First special case: x==0 (derivatives have poles) + if (x_pt.is_zero()) { + const symbol s; + ex ser; + // manually construct the primitive expansion + for (int i=1; i=1 (branch cut) + throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!"); + } + // all other cases should be safe, by now: + throw do_taylor(); // caught by function::series() } @@ -659,16 +1739,16 @@ static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c) if (is_a(m_)) { m = ex_to(m_); } else { - m = lst(m_); + m = lst{m_}; } lst x; if (is_a(x_)) { x = ex_to(x_); } else { - x = lst(x_); + x = lst{x_}; } - c.s << "\\mbox{Li}_{"; - lst::const_iterator itm = m.begin(); + c.s << "\\mathrm{Li}_{"; + auto itm = m.begin(); (*itm).print(c); itm++; for (; itm != m.end(); itm++) { @@ -676,7 +1756,7 @@ static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c) (*itm).print(c); } c.s << "}("; - lst::const_iterator itx = x.begin(); + auto itx = x.begin(); (*itx).print(c); itx++; for (; itx != x.end(); itx++) { @@ -711,7 +1791,7 @@ namespace { // lookup table for special Euler-Zagier-Sums (used for S_n,p(x)) // see fill_Yn() -std::vector > Yn; +std::vector> Yn; int ynsize = 0; // number of Yn[] int ynlength = 100; // initial length of all Yn[i] @@ -732,8 +1812,8 @@ void fill_Yn(int n, const cln::float_format_t& prec) if (n) { std::vector buf(initsize); - std::vector::iterator it = buf.begin(); - std::vector::iterator itprev = Yn[n-1].begin(); + auto it = buf.begin(); + auto itprev = Yn[n-1].begin(); *it = (*itprev) / cln::cl_N(n+1) * one; it++; itprev++; @@ -747,7 +1827,7 @@ void fill_Yn(int n, const cln::float_format_t& prec) Yn.push_back(buf); } else { std::vector buf(initsize); - std::vector::iterator it = buf.begin(); + auto it = buf.begin(); *it = 1 * one; it++; for (int i=2; i<=initsize; i++) { @@ -767,7 +1847,7 @@ 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(); + auto it = Yn[0].begin(); it += ynlength; for (int i=ynlength+1; i<=newsize; i++) { *it = *(it-1) + 1 / cln::cl_N(i) * one; @@ -776,8 +1856,8 @@ void make_Yn_longer(int newsize, const cln::float_format_t& prec) for (int n=1; n::iterator it = Yn[n].begin(); - std::vector::iterator itprev = Yn[n-1].begin(); + auto it = Yn[n].begin(); + auto itprev = Yn[n-1].begin(); it += ynlength; itprev += ynlength; for (int i=ynlength+n+1; i<=newsize+n; i++) { @@ -802,10 +1882,10 @@ 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); } } } @@ -813,23 +1893,23 @@ 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::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::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() + 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::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)); } } @@ -891,10 +1971,20 @@ 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::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) || (n == 0)) { + // the condition abs(1-value)>1 avoids an infinite recursion in the region abs(value)<=1 && abs(value)>0.95 && abs(1-value)<=1 && abs(1-value)>0.95 + // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection + if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) { cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n) * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p); @@ -1006,9 +2098,9 @@ numeric S_num(int n, int p, const numeric& x) cln::cl_N res2; for (int r=0; r 0.95) && (cln::abs(value-9.53) < 9.47)) { + lst m; + m.append(n+1); + for (int s=0; s(res).to_cl_N(); + } else { return S_projection(n, p, value, prec); } @@ -1058,8 +2160,21 @@ numeric S_num(int n, int p, const numeric& x) static ex S_evalf(const ex& n, const ex& p, const ex& x) { - if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a(x)) { - return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(x)); + if (n.info(info_flags::posint) && p.info(info_flags::posint)) { + const int n_ = ex_to(n).to_int(); + const int p_ = ex_to(p).to_int(); + if (is_a(x)) { + const cln::cl_N x_ = ex_to(x).to_cl_N(); + const cln::cl_N result = S_num(n_, p_, x_); + return numeric(result); + } else { + ex x_val = x.evalf(); + if (is_a(x_val)) { + const cln::cl_N x_val_ = ex_to(x_val).to_cl_N(); + const cln::cl_N result = S_num(n_, p_, x_val_); + return numeric(result); + } + } } return S(n, p, x).hold(); } @@ -1072,7 +2187,7 @@ static ex S_eval(const ex& n, const ex& p, const ex& x) return _ex0; } if (x == 1) { - lst m(n+1); + lst m{n+1}; for (int i=ex_to(p).to_int()-1; i>0; i--) { m.append(1); } @@ -1082,7 +2197,11 @@ static ex S_eval(const ex& n, const ex& p, const ex& x) 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)); + int n_ = ex_to(n).to_int(); + int p_ = ex_to(p).to_int(); + const cln::cl_N x_ = ex_to(x).to_cl_N(); + const cln::cl_N result = S_num(n_, p_, x_); + return numeric(result); } } if (n.is_zero()) { @@ -1095,9 +2214,47 @@ static ex S_eval(const ex& n, const ex& p, const ex& x) 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(n, p, x), 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() } @@ -1117,7 +2274,7 @@ static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param) static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c) { - c.s << "\\mbox{S}_{"; + c.s << "\\mathrm{S}_{"; n.print(c); c.s << ","; p.print(c); @@ -1148,7 +2305,7 @@ REGISTER_FUNCTION(S, // anonymous namespace for helper functions namespace { - + // regulates the pole (used by 1/x-transformation) symbol H_polesign("IMSIGN"); @@ -1160,19 +2317,19 @@ 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--) { + for (const auto & it : l) { + 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++) { + } else if (it < -1) { + for (ex count=it+1; count < 0; count++) { mexp.append(0); } mexp.append(-1); } else { - mexp.append(*it); + mexp.append(it); } } @@ -1180,25 +2337,25 @@ bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf) pf = 1; bool has_negative_parameters = false; ex acc = 1; - for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) { - if (*it == 0) { + for (const auto & it : mexp) { + if (it == 0) { acc++; continue; } - if (*it > 0) { - m.append((*it+acc-1) * signum); + if (it > 0) { + m.append((it+acc-1) * signum); } else { - m.append((*it-acc+1) * signum); + m.append((it-acc+1) * signum); } acc = 1; - signum = *it; - pf *= *it; + 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); @@ -1225,9 +2382,9 @@ struct map_trafo_H_convert_to_Li : public map_function if (name == "H") { lst parameter; if (is_a(e.op(0))) { - parameter = ex_to(e.op(0)); + parameter = ex_to(e.op(0)); } else { - parameter = lst(e.op(0)); + parameter = lst{e.op(0)}; } ex arg = e.op(1); @@ -1238,7 +2395,7 @@ 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) || is_a(e)) { return e.map(*this); @@ -1264,9 +2421,9 @@ struct map_trafo_H_convert_to_zeta : public map_function if (name == "H") { lst parameter; if (is_a(e.op(0))) { - parameter = ex_to(e.op(0)); + parameter = ex_to(e.op(0)); } else { - parameter = lst(e.op(0)); + parameter = lst{e.op(0)}; } lst m; @@ -1287,7 +2444,7 @@ struct map_trafo_H_convert_to_zeta : public map_function // remove trailing zeros from H-parameters struct map_trafo_H_reduce_trailing_zeros : public map_function { - ex operator()(const ex& e) + ex operator()(const ex& e) override { if (is_a(e) || is_a(e)) { return e.map(*this); @@ -1299,7 +2456,7 @@ struct map_trafo_H_reduce_trailing_zeros : public map_function if (is_a(e.op(0))) { parameter = ex_to(e.op(0)); } else { - parameter = lst(e.op(0)); + parameter = lst{e.op(0)}; } ex arg = e.op(1); if (parameter.op(parameter.nops()-1) == 0) { @@ -1310,7 +2467,7 @@ struct map_trafo_H_reduce_trailing_zeros : public map_function } // - lst::const_iterator it = parameter.begin(); + auto it = parameter.begin(); while ((it != parameter.end()) && (*it == 0)) { it++; } @@ -1320,7 +2477,7 @@ 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--; } @@ -1372,14 +2529,19 @@ ex convert_H_to_zeta(const lst& m) 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(); - ex signum = _ex1; + auto itm = m.begin(); + auto itx = ++x.begin(); + int signum = 1; pf = _ex1; res.append(*itm); itm++; while (itx != x.end()) { - signum *= *itx; + GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1)); + // XXX: 1 + 0.0*I is considered equal to 1. However the former + // is not automatically converted to a real number. + // Do the conversion explicitly to avoid the + // "numeric::operator>(): complex inequality" exception. + signum *= (*itx != _ex_1) ? 1 : -1; pf *= signum; res.append((*itm) * signum); itm++; @@ -1406,12 +2568,12 @@ ex trafo_H_mult(const ex& h1, const ex& h2) if (h2nops > 1) { hlong = ex_to(h2.op(0)); } else { - hlong = h2.op(0).op(0); + hlong = lst{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)) { return e.map(*this); @@ -1439,7 +2601,7 @@ struct map_trafo_H_mult : public map_function ex result = 1; ex firstH; lst Hlst; - for (int pos=0; pos(e.op(pos)) && is_a(e.op(pos).op(0))) { std::string name = ex_to(e.op(pos).op(0)).get_name(); if (name == "H") { @@ -1473,7 +2635,7 @@ 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") { @@ -1516,7 +2678,38 @@ ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg) 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()); + return e * (-H(lst{ex(0)},1/arg).hold()); + } +} + + +// do integration [ReV] (49) +// put parameter 1 in front of existing parameters +ex trafo_H_prepend_one(const ex& e, const ex& arg) +{ + ex h; + std::string name; + if (is_a(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (std::size_t i=0; i(e.op(i))) { + std::string name = ex_to(e.op(i)).get_name(); + if (name == "H") { + h = e.op(i); + } + } + } + } + if (h != 0) { + lst newparameter = ex_to(h.op(0)); + newparameter.prepend(1); + return e.subs(h == H(newparameter, h.op(1)).hold()); + } else { + return e * H(lst{ex(1)},1-arg).hold(); } } @@ -1533,7 +2726,7 @@ 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") { @@ -1548,8 +2741,8 @@ ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg) ex addzeta = convert_H_to_zeta(newparameter); return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand(); } else { - ex addzeta = convert_H_to_zeta(lst(-1)); - return (e * (addzeta - H(lst(-1),1/arg).hold())).expand(); + ex addzeta = convert_H_to_zeta(lst{ex(-1)}); + return (e * (addzeta - H(lst{ex(-1)},1/arg).hold())).expand(); } } @@ -1566,7 +2759,7 @@ 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") { @@ -1580,7 +2773,7 @@ ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg) 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(); + return (e * H(lst{ex(-1)},(1-arg)/(1+arg)).hold()).expand(); } } @@ -1597,7 +2790,7 @@ 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") { @@ -1611,15 +2804,116 @@ ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg) 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(); + return (e * H(lst{ex(1)},(1-arg)/(1+arg)).hold()).expand(); } } +// do x -> 1-x transformation +struct map_trafo_H_1mx : public map_function +{ + ex operator()(const ex& e) override + { + if (is_a(e) || is_a(e)) { + return e.map(*this); + } + + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + + lst parameter = ex_to(e.op(0)); + ex arg = e.op(1); + + // special cases if all parameters are either 0, 1 or -1 + bool allthesame = true; + if (parameter.op(0) == 0) { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != 0) { + allthesame = false; + break; + } + } + if (allthesame) { + lst newparameter; + for (int i=parameter.nops(); i>0; i--) { + newparameter.append(1); + } + return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold(); + } + } else if (parameter.op(0) == -1) { + throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!"); + } else { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != 1) { + allthesame = false; + break; + } + } + if (allthesame) { + lst newparameter; + for (int i=parameter.nops(); i>0; i--) { + newparameter.append(0); + } + return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold(); + } + } + + lst newparameter = parameter; + newparameter.remove_first(); + + if (parameter.op(0) == 0) { + + // leading zero + ex res = convert_H_to_zeta(parameter); + //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1))); + map_trafo_H_1mx recursion; + ex buffer = recursion(H(newparameter, arg).hold()); + if (is_a(buffer)) { + for (std::size_t i = 0; i < buffer.nops(); i++) { + res -= trafo_H_prepend_one(buffer.op(i), arg); + } + } else { + res -= trafo_H_prepend_one(buffer, arg); + } + return res; + + } else { + + // leading one + map_trafo_H_1mx recursion; + map_trafo_H_mult unify; + ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold(); + std::size_t firstzero = 0; + while (parameter.op(firstzero) == 1) { + firstzero++; + } + for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) { + lst newparameter; + std::size_t j=0; + for (; j<=i; j++) { + newparameter.append(parameter[j+1]); + } + newparameter.append(1); + for (; j 1/x transformation struct map_trafo_H_1overx : public map_function { - ex operator()(const ex& e) + ex operator()(const ex& e) override { if (is_a(e) || is_a(e)) { return e.map(*this); @@ -1635,7 +2929,7 @@ 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 (1-x)/(1+x) transformation struct map_trafo_H_1mxt1px : public map_function { - ex operator()(const ex& e) + ex operator()(const ex& e) override { if (is_a(e) || is_a(e)) { return e.map(*this); @@ -1756,7 +3050,7 @@ struct map_trafo_H_1mxt1px : 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& 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) { @@ -1921,25 +3224,23 @@ static ex H_evalf(const ex& x1, const ex& x2) } // ... 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--) { + for (const auto & it : morg) { + 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++) { + } else if (it <= -1) { + for (ex count=it+1; count < 0; count++) { m.append(0); } m.append(-1); } else { - m.append(*it); + 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; @@ -1948,7 +3249,7 @@ static ex H_evalf(const ex& x1, const ex& x2) // 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(); + for (auto 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()); @@ -1962,19 +3263,20 @@ static ex H_evalf(const ex& x1, const ex& x2) 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()); + for (const auto & it : m_lst) { + m_int.push_back(ex_to(it).to_int()); } return numeric(H_do_sum(m_int, x)); } } + symbol xtemp("xtemp"); ex res = 1; // ensure that the realpart of the argument is positive if (cln::realpart(x) < 0) { x = -x; - for (int i=0; i (1-x)/(1+x) - map_trafo_H_1mxt1px trafo; - res *= trafo(H(m, xtemp)); - } else { - // x -> 1/x + // x -> 1/x + if (cln::abs(x) >= 2.0) { map_trafo_H_1overx trafo; - res *= trafo(H(m, xtemp)); + res *= trafo(H(m, xtemp).hold()); if (cln::imagpart(x) <= 0) { res = res.subs(H_polesign == -I*Pi); } else { res = res.subs(H_polesign == I*Pi); } + return res.subs(xtemp == numeric(x)).evalf(); + } + + // check for letters (-1) + bool has_minus_one = false; + for (const auto & it : m) { + if (it == -1) + has_minus_one = true; } - // 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); + // check transformations for 0.95 <= |x| < 2.0 + + // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47 + if (cln::abs(x-9.53) <= 9.47) { + // x -> (1-x)/(1+x) + map_trafo_H_1mxt1px trafo; + res *= trafo(H(m, xtemp).hold()); + } else { + // x -> 1-x + if (has_minus_one) { + map_trafo_H_convert_to_Li filter; + return filter(H(m, numeric(x)).hold()).evalf(); + } + map_trafo_H_1mx trafo; + res *= trafo(H(m, xtemp).hold()); + } return res.subs(xtemp == numeric(x)).evalf(); } @@ -2022,7 +3333,7 @@ static ex H_eval(const ex& m_, const ex& x) if (is_a(m_)) { m = ex_to(m_); } else { - m = lst(m_); + m = lst{m_}; } if (m.nops() == 0) { return _ex1; @@ -2051,8 +3362,8 @@ static ex H_eval(const ex& m_, const ex& x) pos1 = *m.begin(); p = _ex1; } - for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) { - if ((*it).info(info_flags::integer)) { + for (auto it = ++m.begin(); it != m.end(); it++) { + if (it->info(info_flags::integer)) { if (step == 0) { if (*it > _ex1) { if (pos1 == _ex0) { @@ -2133,9 +3444,8 @@ static ex H_eval(const ex& m_, const ex& x) static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options) { - epvector seq; - seq.push_back(expair(H(m, x), 0)); - return pseries(rel, seq); + epvector seq { expair(H(m, x), 0) }; + return pseries(rel, std::move(seq)); } @@ -2149,7 +3459,7 @@ static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param) if (is_a(m_)) { m = ex_to(m_); } else { - m = lst(m_); + m = lst{m_}; } ex mb = *m.begin(); if (mb > _ex1) { @@ -2177,10 +3487,10 @@ static void H_print_latex(const ex& m_, const ex& x, const print_context& c) if (is_a(m_)) { m = ex_to(m_); } else { - m = lst(m_); + m = lst{m_}; } - c.s << "\\mbox{H}_{"; - lst::const_iterator itm = m.begin(); + c.s << "\\mathrm{H}_{"; + auto itm = m.begin(); (*itm).print(c); itm++; for (; itm != m.end(); itm++) { @@ -2210,7 +3520,7 @@ ex convert_H_to_Li(const ex& m, const ex& x) if (is_a(m)) { return filter2(filter(H(m, x).hold())); } else { - return filter2(filter(H(lst(m), x).hold())); + return filter2(filter(H(lst{m}, x).hold())); } } @@ -2230,13 +3540,6 @@ 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) { @@ -2251,44 +3554,39 @@ 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 < (int)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); @@ -2300,20 +3598,18 @@ cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk) factor = factor * lambda; N++; res = res + crX[N] * factor / (N+Sqk); - } while ((res != resbuf) || cln::zerop(crX[N])); + } while (((res != resbuf) || cln::zerop(crX[N])) && (N+1 < crX.size())); return res; } // [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(); + auto it = f_kj.begin(); cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); t0 = cln::exp(-lambda); @@ -2336,7 +3632,8 @@ 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(); @@ -2348,7 +3645,7 @@ cln::cl_N crandall_Z(const std::vector& s) t0buf = t0; q++; t0 = t0 + f_kj[q+j-2][s[0]-1]; - } while (t0 != t0buf); + } while ((t0 != t0buf) && (q+j-1 < f_kj.size())); return t0 / cln::factorial(s[0]-1); } @@ -2365,7 +3662,7 @@ cln::cl_N crandall_Z(const std::vector& s) t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]); } t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1]; - } while (t[0] != t0buf); + } while ((t[0] != t0buf) && (q+j-1 < f_kj.size())); return t[0] / cln::factorial(s[0]-1); } @@ -2377,6 +3674,8 @@ 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; @@ -2384,6 +3683,7 @@ 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; @@ -2395,8 +3695,11 @@ cln::cl_N zeta_do_sum_Crandall(const std::vector& s) L2 = 511; } else if (Digits < 808) { L2 = 1023; - } else { + } else if (Digits < 1636) { L2 = 2047; + } else { + // [Cra] section 6, log10(lambda/2/Pi) approx -0.79 for lambda=319/320, add some extra digits + L2 = std::pow(2, ceil( std::log2((long(Digits))/0.79 + 40 )) ) - 1; } cln::cl_N res; @@ -2410,7 +3713,8 @@ 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); @@ -2424,12 +3728,13 @@ 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; } @@ -2487,7 +3794,7 @@ 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(x); std::vector r(count); + std::vector si(count); // check parameters and convert them - lst::const_iterator it1 = xlst.begin(); - std::vector::iterator it2 = r.begin(); + auto it1 = xlst.begin(); + auto it2 = r.begin(); + auto it_swrite = si.begin(); do { if (!(*it1).info(info_flags::posint)) { return zeta(x).hold(); } *it2 = ex_to(*it1).to_int(); + *it_swrite = 1; it1++; it2++; + it_swrite++; } while (it2 != r.end()); // check for divergence @@ -2598,6 +3909,10 @@ static ex zeta1_evalf(const ex& x) return zeta(x).hold(); } + // use Hoelder convolution if Digits is large + if (Digits>50) + return numeric(zeta_do_Hoelder_convolution(r, si)); + // decide on summation algorithm // this is still a bit clumsy int limit = (Digits>17) ? 10 : 6; @@ -2635,18 +3950,18 @@ static ex zeta1_eval(const ex& m) if (y.is_zero()) { return _ex_1_2; } - if (y.is_equal(_num1)) { + if (y.is_equal(*_num1_p)) { return zeta(m).hold(); } if (y.info(info_flags::posint)) { if (y.info(info_flags::odd)) { return zeta(m).hold(); } else { - return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, 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; } @@ -2678,7 +3993,7 @@ 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(); + auto it = m.begin(); (*it).print(c); it++; for (; it != m.end(); it++) { @@ -2722,10 +4037,10 @@ static ex zeta2_evalf(const ex& x, const ex& s) std::vector si(count); // check parameters and convert them - lst::const_iterator it_xread = xlst.begin(); - lst::const_iterator it_sread = slst.begin(); - std::vector::iterator it_xwrite = xi.begin(); - std::vector::iterator it_swrite = si.begin(); + auto it_xread = xlst.begin(); + auto it_sread = slst.begin(); + auto it_xwrite = xi.begin(); + auto it_swrite = si.begin(); do { if (!(*it_xread).info(info_flags::posint)) { return zeta(x, s).hold(); @@ -2759,8 +4074,8 @@ static ex zeta2_eval(const ex& m, const ex& s_) { 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)) { + for (const auto & it : s) { + if (it.info(info_flags::positive)) { continue; } return zeta(m, s_).hold(); @@ -2795,17 +4110,17 @@ static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c if (is_a(m_)) { m = ex_to(m_); } else { - m = lst(m_); + m = lst{m_}; } lst s; if (is_a(s_)) { s = ex_to(s_); } else { - s = lst(s_); + s = lst{s_}; } c.s << "\\zeta("; - lst::const_iterator itm = m.begin(); - lst::const_iterator its = s.begin(); + auto itm = m.begin(); + auto its = s.begin(); if (*its < 0) { c.s << "\\overline{"; (*itm).print(c);