X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_nstdsums.cpp;h=67b06bae54f605599aa09d5acbb06352d9f714c5;hp=88f0fa4ee5f313d3a3c2925294616984782c94c4;hb=11c0c610d6f47476b82f3ab118b0be37ed3ef747;hpb=270969d036bd27a8454442501f8eb241fa66c9b2 diff --git a/ginac/inifcns_nstdsums.cpp b/ginac/inifcns_nstdsums.cpp index 88f0fa4e..67b06bae 100644 --- a/ginac/inifcns_nstdsums.cpp +++ b/ginac/inifcns_nstdsums.cpp @@ -4,12 +4,12 @@ * * The functions are: * classical polylogarithm Li(n,x) - * 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) + * 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)) + * 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: * @@ -25,7 +25,7 @@ * 0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single * number --- notation. * - * - All functions can be nummerically evaluated with arguments in the whole complex plane. The parameters + * - 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. * @@ -47,7 +47,7 @@ */ /* - * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2016 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 @@ -64,11 +64,6 @@ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ -#include -#include -#include -#include - #include "inifcns.h" #include "add.h" @@ -84,6 +79,10 @@ #include "utils.h" #include "wildcard.h" +#include +#include +#include +#include namespace GiNaC { @@ -103,7 +102,7 @@ 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; @@ -125,7 +124,7 @@ 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++; @@ -150,7 +149,7 @@ 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++; @@ -174,7 +173,7 @@ 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++; @@ -210,7 +209,7 @@ void double_Xn() } } // X_n - for (int n=2; n 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); } @@ -363,15 +366,17 @@ 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(x).to_cl_N(); - cln::cl_N result = -cln::expt(cln::log(x_), n-1) * cln::log(1-x_) / cln::factorial(n-1); + if (cln::abs(realpart(x)) < 0.4 && cln::abs(cln::abs(x)-1) < 0.01) { + 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) @@ -441,7 +444,7 @@ numeric Lin_numeric(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)); @@ -480,51 +489,33 @@ 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]; } -// converts parameter types and calls multipleLi_do_sum (convenience function for G_numeric) -cln::cl_N mLi_do_summation(const lst& m, const lst& x) -{ - std::vector m_int; - std::vector x_cln; - for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { - m_int.push_back(ex_to(*itm).to_int()); - x_cln.push_back(ex_to(*itx).to_cl_N()); - } - return multipleLi_do_sum(m_int, x_cln); -} - - // forward declaration for Li_eval() lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf); -// holding dummy-symbols for the G/Li transformations -std::vector gsyms; - - // type used by the transformation functions for G typedef std::vector Gparameter; // G_eval1-function for G transformations -ex G_eval1(int a, int scale) +ex G_eval1(int a, int scale, const exvector& gsyms) { if (a != 0) { const ex& scs = gsyms[std::abs(scale)]; @@ -541,7 +532,7 @@ ex G_eval1(int a, int scale) // G_eval-function for G transformations -ex G_eval(const Gparameter& a, int scale) +ex G_eval(const Gparameter& a, int scale, const exvector& gsyms) { // check for properties of G ex sc = gsyms[std::abs(scale)]; @@ -549,9 +540,9 @@ ex G_eval(const Gparameter& a, int scale) 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)]; + for (const auto & it : a) { + if (it != 0) { + const ex sym = gsyms[std::abs(it)]; newa.append(sym); all_zero = false; if (sym != sc) { @@ -569,38 +560,25 @@ ex G_eval(const Gparameter& a, int scale) // later on in the transformation if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) { // do shuffle - Gparameter short_a; - Gparameter::const_iterator it = a.begin(); - ++it; - for (; it != a.end(); ++it) { - short_a.push_back(*it); - } - ex result = G_eval1(a.front(), scale) * G_eval(short_a, scale); - it = short_a.begin(); - for (int i=1; i G({1};y)^k / k! if (all_ones && a.size() > 1) { - return pow(G_eval1(a.front(),scale), count_ones) / factorial(count_ones); + return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones); } // G({0,...,0};y) -> log(y)^k / k! @@ -613,9 +591,9 @@ ex G_eval(const Gparameter& a, int scale) 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)]; + for (const auto & it : a) { + if (it != 0) { + const ex& sym = gsyms[std::abs(it)]; x.append(argbuf / sym); m.append(mval); mval = _ex1; @@ -651,14 +629,14 @@ Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals) // 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) + 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) { + auto lastnonzero = a.end(); + for (auto it = a.begin(); it != a.end(); ++it) { if (std::abs(*it) > 0) { ++depth; trailing_zeros = 0; @@ -673,6 +651,8 @@ Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale, ++trailing_zeros; } } + if (lastnonzero == a.end()) + return a.end(); return ++lastnonzero; } @@ -693,7 +673,7 @@ Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int sc // handles trailing zeroes for an otherwise convergent integral -ex trailing_zeros_G(const Gparameter& a, int scale) +ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms) { bool convergent; int depth, trailing_zeros; @@ -705,23 +685,23 @@ ex trailing_zeros_G(const Gparameter& a, int scale) if ((trailing_zeros > 0) && (depth > 0)) { ex result; Gparameter new_a(a.begin(), a.end()-1); - result += G_eval1(0, scale) * trailing_zeros_G(new_a, scale); - for (Gparameter::const_iterator it = a.begin(); it != last; ++it) { + 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); + result -= trailing_zeros_G(new_a, scale, gsyms); } return result / trailing_zeros; } else { - return G_eval(a, scale); + 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) +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 ) @@ -752,15 +732,21 @@ ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, i result -= I*Pi; } if (psize) { - result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front()); + 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()); + 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()); + result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), + new_pending_integrals.front(), + gsyms); return result; } @@ -772,14 +758,16 @@ ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, i //term zeta_m result -= zeta(a.size()); if (psize) { - result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front()); + 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); + 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 ) @@ -787,10 +775,12 @@ ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, i new_pending_integrals_2.push_back(scale); new_pending_integrals_2.push_back(0); if (psize) { - result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front()) - * depth_one_trafo_G(new_pending_integrals_2, new_a, scale); + 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); + result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms); } return result; @@ -799,11 +789,13 @@ ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, i // forward declaration ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2, - const Gparameter& pendint, const Gparameter& a_old, int scale); + 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) +ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale, + const exvector& gsyms, bool flag_trailing_zeros_only) { // main recursion routine // @@ -818,21 +810,22 @@ ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale) 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(); + 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; + result = 1; } else { - result = G_eval(a, scale); + result = G_eval(a, scale, gsyms); } if (pendint.size() > 0) { - result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front()); + result *= trailing_zeros_G(convert_pending_integrals_G(pendint), + pendint.front(), + gsyms); } return result; } @@ -841,28 +834,30 @@ ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale) if (trailing_zeros > 0) { ex result; Gparameter new_a(a.begin(), a.end()-1); - result += G_eval1(0, scale) * G_transform(pendint, new_a, scale); - for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) { + 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); + result -= G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only); } return result / trailing_zeros; } - // convergence case - if (convergent) { + // convergence case or flag_trailing_zeros_only + if (convergent || flag_trailing_zeros_only) { if (pendint.size() > 0) { - return G_eval(convert_pending_integrals_G(pendint), pendint.front()) * G_eval(a, scale); + return G_eval(convert_pending_integrals_G(pendint), + pendint.front(), gsyms) * + G_eval(a, scale, gsyms); } else { - return G_eval(a, scale); + return G_eval(a, scale, gsyms); } } // call basic transformation for depth equal one if (depth == 1) { - return depth_one_trafo_G(pendint, a, scale); + return depth_one_trafo_G(pendint, a, scale, gsyms); } // do recursion @@ -877,9 +872,10 @@ ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale) Gparameter a1(a.begin(),min_it+1); Gparameter a2(min_it+1,a.end()); - ex result = G_transform(pendint,a2,scale)*G_transform(empty,a1,scale); + 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); + result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms, flag_trailing_zeros_only); return result; } @@ -890,9 +886,10 @@ ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale) 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); + 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()); + result *= trailing_zeros_G(convert_pending_integrals_G(pendint), + pendint.front(), gsyms); } // other terms @@ -901,29 +898,33 @@ ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale) if (changeit != new_a.begin()) { // smallest in the middle new_pendint.push_back(*changeit); - result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front()) - * G_transform(empty, new_a, scale); + 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); + 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()) - * G_transform(empty, new_a, 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); *changeit = *min_it; - result -= G_transform(new_pendint, new_a, scale); + 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()) - * G_transform(empty, new_a, 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()) - * G_transform(empty, new_a, 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); *changeit = *min_it; - result += G_transform(new_pendint, new_a, scale); + result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only); } return result; } @@ -932,27 +933,28 @@ ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale) // 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 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); + 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); + 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); + return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only); } Gparameter a1_removed(a1.begin()+1,a1.end()); @@ -964,203 +966,258 @@ ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2 a01.push_back( a1[0] ); a02.push_back( a2[0] ); - return shuffle_G(a01,a1_removed,a2,pendint,a_old,scale) - + shuffle_G(a02,a1,a2_removed,pendint,a_old,scale); + 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 -ex G_numeric(const lst& x, const lst& s, const ex& y) +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) { - // check for convergence and necessary accelerations - bool need_trafo = false; - bool need_hoelder = false; - int depth = 0; - for (lst::const_iterator it = x.begin(); it != x.end(); ++it) { - if (!(*it).is_zero()) { - ++depth; - if (abs(*it) - y < -pow(10,-Digits+2)) { - need_trafo = true; - break; + 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; + } } - if (abs((abs(*it) - y)/y) < 0.01) { - need_hoelder = true; + } while (adjustp); + cln::cl_RA q = p/(p-1); + std::vector qlstx; + std::vector qlsts; + for (std::size_t j = r; j >= 1; --j) { + qlstx.push_back(cln::cl_N(1) - x[j-1]); + if (instanceof(x[j-1], cln::cl_R_ring) && realpart(x[j-1]) > 1) { + qlsts.push_back(1); + } else { + qlsts.push_back(-s[j-1]); } } + if (qlstx.size() > 0) { + buffer = buffer*G_numeric(qlstx, qlsts, 1/q); + } + std::vector plstx; + std::vector plsts; + for (std::size_t j = r+1; j <= size; ++j) { + plstx.push_back(x[j-1]); + plsts.push_back(s[j-1]); + } + if (plstx.size() > 0) { + buffer = buffer*G_numeric(plstx, plsts, 1/p); + } + result = result + buffer; } - if (x.op(x.nops()-1).is_zero()) { - need_trafo = true; - } - if (depth == 1 && !need_trafo) { - return -Li(x.nops(), y / x.op(x.nops()-1)).evalf(); + 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 - if (need_trafo) { - - // sort (|x|<->position) to determine indices - std::multimap sortmap; - int size = 0; - for (int i=0; i(abs(x[i]), i)); - ++size; - } - } - // include upper limit (scale) - sortmap.insert(std::pair(abs(y), x.nops())); - - // generate missing dummy-symbols - int i = 1; - gsyms.clear(); - gsyms.push_back(symbol("GSYMS_ERROR")); - ex lastentry; - for (std::multimap::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) { - if (it != sortmap.begin()) { - if (it->second < x.nops()) { - if (x[it->second] == lastentry) { - gsyms.push_back(gsyms.back()); - continue; - } - } else { - if (y == lastentry) { - gsyms.push_back(gsyms.back()); - continue; - } +}; + + +// 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; } - } - std::ostringstream os; - os << "a" << i; - gsyms.push_back(symbol(os.str())); - ++i; - if (it->second < x.nops()) { - lastentry = x[it->second]; } else { - lastentry = y; + 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.nops()); - lst subslst; - int pos = 1; - int scale; - for (std::multimap::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) { - if (it->second < x.nops()) { - if (s[it->second] > 0) { - a[it->second] = pos; - } else { - a[it->second] = -pos; - } - subslst.append(gsyms[pos] == x[it->second]); + // 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 { - scale = pos; - subslst.append(gsyms[pos] == y); + a[it->second] = -int(pos); } - ++pos; + subslst[gsyms[pos]] = numeric(x[it->second]); + } else { + scale = pos; + subslst[gsyms[pos]] = numeric(y); } - - // do transformation - Gparameter pendint; - ex result = G_transform(pendint, a, scale); - // replace dummy symbols with their values - result = result.eval().expand(); - result = result.subs(subslst).evalf(); - - return result; + ++pos; } - // do acceleration transformation (hoelder convolution [BBB]) - if (need_hoelder) { - - ex result; - const int size = x.nops(); - lst newx; - for (lst::const_iterator it = x.begin(); it != x.end(); ++it) { - newx.append(*it / y); - } - - for (int r=0; r<=size; ++r) { - ex buffer = pow(-1, r); - ex p = 2; - bool adjustp; - do { - adjustp = false; - for (lst::const_iterator it = newx.begin(); it != newx.end(); ++it) { - if (*it == 1/p) { - p += (3-p)/2; - adjustp = true; - continue; - } - } - } while (adjustp); - ex q = p / (p-1); - lst qlstx; - lst qlsts; - for (int j=r; j>=1; --j) { - qlstx.append(1-newx.op(j-1)); - if (newx.op(j-1).info(info_flags::real) && newx.op(j-1) > 1 && newx.op(j-1) <= 2) { - qlsts.append( s.op(j-1)); - } else { - qlsts.append( -s.op(j-1)); - } - } - if (qlstx.nops() > 0) { - buffer *= G_numeric(qlstx, qlsts, 1/q); - } - lst plstx; - lst plsts; - for (int j=r+1; j<=size; ++j) { - plstx.append(newx.op(j-1)); - plsts.append(s.op(j-1)); - } - if (plstx.nops() > 0) { - buffer *= G_numeric(plstx, plsts, 1/p); - } - result += buffer; + // 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; } - return result; } + 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 - lst newx; - lst m; + std::vector newx; + newx.reserve(x.size()); + std::vector m; + m.reserve(x.size()); int mcount = 1; - ex sign = 1; - ex factor = y; - for (lst::const_iterator it = x.begin(); it != x.end(); ++it) { - if ((*it).is_zero()) { + int sign = 1; + cln::cl_N factor = y; + for (auto & xi : x) { + if (zerop(xi)) { ++mcount; } else { - newx.append(factor / (*it)); - factor = *it; - m.append(mcount); + newx.push_back(factor/xi); + factor = xi; + m.push_back(mcount); mcount = 1; sign = -sign; } } - return sign * numeric(mLi_do_summation(m, newx)); + return sign*multipleLi_do_sum(m, newx); } ex mLi_numeric(const lst& m, const lst& x) { // let G_numeric do the transformation - lst newx; - lst s; - ex factor = 1; - for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { + 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.append(0); - s.append(1); + 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); } - newx.append(factor / *itx); - factor /= *itx; - s.append(1); } - return pow(-1, m.nops()) * G_numeric(newx, s, _ex1); + return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1))); } @@ -1178,31 +1235,42 @@ ex mLi_numeric(const lst& m, const lst& x) static ex G2_evalf(const ex& x_, const ex& y) { - if (!y.info(info_flags::positive)) { + 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_); + lst x = is_a(x_) ? ex_to(x_) : lst{x_}; if (x.nops() == 0) { return _ex1; } if (x.op(0) == y) { return G(x_, y).hold(); } - lst s; + 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)) { + for (const auto & it : x) { + if (!it.info(info_flags::numeric)) { return G(x_, y).hold(); } - if (*it != _ex0) { + if (it != _ex0) { all_zero = false; } - s.append(+1); + 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()); } - return G_numeric(x, s, y); + 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); } @@ -1210,30 +1278,36 @@ 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)) { + 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_); + lst x = is_a(x_) ? ex_to(x_) : lst{x_}; if (x.nops() == 0) { return _ex1; } if (x.op(0) == y) { return G(x_, y).hold(); } - lst s; + 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)) { + for (const auto & it : x) { + if (!it.info(info_flags::numeric)) { return G(x_, y).hold(); } - if (!(*it).info(info_flags::crational)) { + if (!it.info(info_flags::crational)) { crational = false; } - if (*it != _ex0) { + if (it != _ex0) { all_zero = false; } - s.append(+1); + 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()); @@ -1244,14 +1318,19 @@ static ex G2_eval(const ex& x_, const ex& y) if (crational) { return G(x_, y).hold(); } - return G_numeric(x, s, y); + 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). - do_not_evalf_params(). overloaded(2)); //TODO // derivative_func(G2_deriv). @@ -1260,11 +1339,11 @@ unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2). static ex G3_evalf(const ex& x_, const ex& s_, const ex& y) { - if (!y.info(info_flags::positive)) { + 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_); + 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(); } @@ -1274,9 +1353,10 @@ static ex G3_evalf(const ex& x_, const ex& s_, const ex& y) if (x.op(0) == y) { return G(x_, s_, y).hold(); } - lst sn; + 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) { + for (auto itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) { if (!(*itx).info(info_flags::numeric)) { return G(x_, y).hold(); } @@ -1286,16 +1366,36 @@ static ex G3_evalf(const ex& x_, const ex& s_, const ex& y) if (*itx != _ex0) { all_zero = false; } - if (*its >= 0) { - sn.append(+1); - } else { - sn.append(-1); + 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()); } - return G_numeric(x, sn, y); + 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); } @@ -1303,11 +1403,11 @@ 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)) { + 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_); + 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(); } @@ -1317,10 +1417,11 @@ static ex G3_eval(const ex& x_, const ex& s_, const ex& y) if (x.op(0) == y) { return G(x_, s_, y).hold(); } - lst sn; + 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) { + 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(); } @@ -1333,10 +1434,25 @@ static ex G3_eval(const ex& x_, const ex& s_, const ex& y) if (*itx != _ex0) { all_zero = false; } - if (*its >= 0) { - sn.append(+1); - } else { - sn.append(-1); + 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) { @@ -1348,14 +1464,21 @@ static ex G3_eval(const ex& x_, const ex& s_, const ex& y) if (crational) { return G(x_, s_, y).hold(); } - return G_numeric(x, sn, y); + 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); } +// 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). - do_not_evalf_params(). overloaded(2)); //TODO // derivative_func(G3_deriv). @@ -1376,12 +1499,18 @@ static ex Li_evalf(const ex& m_, const ex& x_) // classical polylogs if (m_.info(info_flags::posint)) { if (x_.info(info_flags::numeric)) { - return Lin_numeric(ex_to(m_).to_int(), ex_to(x_)); + 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)) { - return Lin_numeric(ex_to(m_).to_int(), ex_to(x_val)); + 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); } } } @@ -1400,7 +1529,7 @@ static ex Li_evalf(const ex& m_, const ex& x_) return Li(m_,x_).hold(); } - for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { + for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { if (!(*itm).info(info_flags::posint)) { return Li(m_, x_).hold(); } @@ -1436,7 +1565,7 @@ static ex Li_eval(const ex& m_, const ex& x_) 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) { + for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { if (!(*itm).info(info_flags::posint)) { return Li(m_,x_).hold(); } @@ -1457,7 +1586,17 @@ static ex Li_eval(const ex& m_, const ex& x_) } } if (is_zeta) { - return zeta(m_,x_); + 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 (is_H) { ex prefactor; @@ -1495,7 +1634,10 @@ static ex Li_eval(const ex& m_, const ex& x_) } } if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) { - return Lin_numeric(ex_to(m_).to_int(), ex_to(x_)); + 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(); @@ -1504,9 +1646,35 @@ static ex Li_eval(const ex& m_, const ex& x_) 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() } @@ -1545,16 +1713,16 @@ 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++) { @@ -1562,7 +1730,7 @@ 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++) { @@ -1597,7 +1765,7 @@ 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] @@ -1618,8 +1786,8 @@ 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++; @@ -1633,7 +1801,7 @@ 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++) { @@ -1653,7 +1821,7 @@ 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; @@ -1662,8 +1830,8 @@ 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++) { @@ -1688,10 +1856,10 @@ 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); } } } @@ -1699,23 +1867,23 @@ 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)); } } @@ -1777,10 +1945,20 @@ 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); @@ -1892,9 +2072,9 @@ 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); } @@ -1945,12 +2135,18 @@ 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)) { + const int n_ = ex_to(n).to_int(); + const int p_ = ex_to(p).to_int(); if (is_a(x)) { - return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(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)) { - return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(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); } } } @@ -1965,7 +2161,7 @@ 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); } @@ -1975,7 +2171,11 @@ 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()) { @@ -1988,9 +2188,47 @@ 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() } @@ -2010,7 +2248,7 @@ 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); @@ -2041,7 +2279,7 @@ REGISTER_FUNCTION(S, // anonymous namespace for helper functions namespace { - + // regulates the pole (used by 1/x-transformation) symbol H_polesign("IMSIGN"); @@ -2053,19 +2291,19 @@ 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); } } @@ -2073,25 +2311,25 @@ 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); @@ -2118,9 +2356,9 @@ 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); @@ -2131,7 +2369,7 @@ 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); @@ -2157,9 +2395,9 @@ 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; @@ -2180,7 +2418,7 @@ 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); @@ -2192,7 +2430,7 @@ 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) { @@ -2203,7 +2441,7 @@ 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++; } @@ -2213,7 +2451,7 @@ 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--; } @@ -2265,14 +2503,19 @@ 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(); + auto itm = m.begin(); + auto itx = ++x.begin(); int signum = 1; pf = _ex1; res.append(*itm); itm++; while (itx != x.end()) { - signum *= (*itx > 0) ? 1 : -1; + 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++; @@ -2299,12 +2542,12 @@ 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); @@ -2332,7 +2575,7 @@ 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") { @@ -2366,7 +2609,7 @@ 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") { @@ -2409,7 +2652,38 @@ 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(); } } @@ -2426,7 +2700,7 @@ 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") { @@ -2441,8 +2715,8 @@ 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(); } } @@ -2459,7 +2733,7 @@ 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") { @@ -2473,7 +2747,7 @@ 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(); } } @@ -2490,7 +2764,7 @@ 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") { @@ -2504,15 +2778,116 @@ 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); @@ -2528,7 +2903,7 @@ 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); @@ -2649,7 +3024,7 @@ 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 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); + has_minus_one = true; } else { - m.append(*it); + m.append(it); } } - // check for the applicability of transformations - // - // first condition: since the transformations produce a lot of terms, - // they are only efficient for argument near the boundary |x| = 1, then no - // transformation is needed - // second condition: veto for region around +-I to avoid endless recursion - // with the (1-x)/(1+x) transformation. 1.198 is sqrt(1.4142) is the - // boundary of the problematic transformation. - // - if (cln::abs(x) < 0.95 || (cln::abs(x) < 1 && cln::abs(x-1) >= 1.198)) { + // do summation + if (cln::abs(x) < 0.95) { lst m_lst; lst s_lst; ex pf; @@ -2856,7 +3225,7 @@ 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()); @@ -2870,19 +3239,20 @@ 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 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()); } - - // 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(); } @@ -2930,7 +3302,7 @@ 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; @@ -2959,8 +3331,8 @@ 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) { @@ -3041,9 +3413,8 @@ 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)); } @@ -3057,7 +3428,7 @@ 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) { @@ -3085,10 +3456,10 @@ 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++) { @@ -3118,7 +3489,7 @@ 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())); } } @@ -3138,13 +3509,6 @@ 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) { @@ -3159,44 +3523,39 @@ 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); @@ -3214,14 +3573,12 @@ 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(); + auto it = f_kj.begin(); cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); t0 = cln::exp(-lambda); @@ -3244,7 +3601,8 @@ 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(); @@ -3285,6 +3643,8 @@ 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; @@ -3292,6 +3652,7 @@ 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; @@ -3318,7 +3679,8 @@ 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); @@ -3332,12 +3694,13 @@ 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; } @@ -3395,7 +3760,7 @@ 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 r(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(); do { if (!(*it1).info(info_flags::posint)) { return zeta(x).hold(); @@ -3586,7 +3951,7 @@ 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++) { @@ -3630,10 +3995,10 @@ 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(); @@ -3667,8 +4032,8 @@ 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(); @@ -3703,17 +4068,17 @@ 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);