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Better precision in divided differences #404

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Jan 12, 2021
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Better precision in divided differences #404

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7 changes: 5 additions & 2 deletions src/Smearing.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -42,8 +42,11 @@ function occupation_divided_difference(S::SmearingFunction, x, y, εF, temperatu
end
end
function divided_difference_(f, fder, x, y)
# This is only accurate to sqrt(ε)
abs(x-y) < sqrt(eps(typeof(x))) && return fder((x+y)/2)
# (f(x) - f(y))/(x - y) is accurate to ε/|x-y|
# so for x ~= y we use the approximation (f'(x)+f'(y))/2,
# which is accurate to |x-y|^2, and therefore better when |x-y| ≤ cbrt(ε)
# The resulting method is accurate to ε^2/3
abs(x-y) < cbrt(eps(typeof(x))) && return (fder(x) + fder(y))/2
(f(x)-f(y)) / (x-y)
end

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