Log-Arcsine Algorithm

[Albert Chan]

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

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

0 ENTER 1
R/S
R/S
R/S
…

1.41421356237
1.53073372946
1.56072257613
1.56827424527
1.57016557848
1.57063862547
1.57075690057
1.57078647018
1.57079386264
...

Above initial conditions for θ = asin(x):

s_-1 = x*√(1-x*x) = sin(θ)*cos(θ) = sin(2θ)/2
s₀  = x = sin(θ)

The program then iterate for s₁ = 2*sin(θ/2), s₂ = 4*sin(θ/4), ... --> s_∞ = θ = asin(x)

Sequence is identical to asin(x) half-angle formula. Note: asinq(x) = asin(√x)

(03-31-2022 11:07 PM)Albert Chan Wrote:  asinq(x) = 2 * asinq(x/2/(1+sqrt(1-x)))

>>> x = 1 # iterate for asin(√1)
>>> for k in range(1,20):
...            x /= 2*(1+sqrt(1-x))
...            print 2**k * sqrt(x)
...
1.41421356237
1.53073372946
1.56072257613
1.56827424527
1.57016557848
1.57063862547
1.57075690057
1.57078647018
1.57079386264
...