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