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File: cohen_acceleration.hpp
// (C) Copyright Nick Thompson 2020. // Use, modification and distribution are subject to the // Boost Software License, Version 1.0. (See accompanying file // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) #ifndef BOOST_MATH_TOOLS_COHEN_ACCELERATION_HPP #define BOOST_MATH_TOOLS_COHEN_ACCELERATION_HPP #include <limits> #include <cmath> #include <cstdint> namespace boost::math::tools { // Algorithm 1 of https://people.mpim-bonn.mpg.de/zagier/files/exp-math-9/fulltext.pdf // Convergence Acceleration of Alternating Series: Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier template<class G> auto cohen_acceleration(G& generator, std::int64_t n = -1) { using Real = decltype(generator()); // This test doesn't pass for float128, sad! //static_assert(std::is_floating_point_v<Real>, "Real must be a floating point type."); using std::log; using std::pow; using std::ceil; using std::sqrt; auto n_ = static_cast<Real>(n); if (n < 0) { // relative error grows as 2*5.828^-n; take 5.828^-n < eps/4 => -nln(5.828) < ln(eps/4) => n > ln(4/eps)/ln(5.828). // Is there a way to do it rapidly with std::log2? (Yes, of course; but for primitive types it's computed at compile-time anyway.) n_ = static_cast<Real>(ceil(log(4/std::numeric_limits<Real>::epsilon())*0.5672963285532555)); n = static_cast<std::int64_t>(n_); } // d can get huge and overflow if you pick n too large: auto d = static_cast<Real>(pow(3 + sqrt(Real(8)), n)); d = (d + 1/d)/2; Real b = -1; Real c = -d; Real s = 0; for (Real k = 0; k < n_; ++k) { c = b - c; s += c*generator(); b = (k+n_)*(k-n_)*b/((k+Real(1)/Real(2))*(k+1)); } return s/d; } } #endif