Log-Arcsine Algorithm

[Albert Chan]

(11-02-2022 10:22 PM)Thomas Klemm Wrote:  The following recurrence relation is used:

\(
\begin{align}
s_{n+1} = s_n \sqrt{\frac{2 s_n}{s_n + s_{n-1}}}
\end{align}
\)

If we let y = ln(x), and

s_-1 = (x² - 1/x²)/4 = sinh(2y)/2
s₀ = (x - 1/x)/2 = sinh(y)

then

s₁ = 2*sinh(y/2)
s₂ = 4*sinh(y/4) ... --> s_∞ = y = ln(x),

sin(θ) = sinh(θ*i)/i, this explained why Log-Arcsine Algorithm have same recurrence relation.