numerical methods with triple exp convergeance ?

[tommy1729]

Are there numerical methods with triple exp convergeance ?

Say we want sqrt(2) by iterations.

iterating rational functions is a convergeance speed of double exponential speed at best.

Like newtons method is of speed a^(2^n) and b iterations of it are of speed a^(2^(bn)) what is basically the same ofcourse.

In fact all iterations of analytic functions ( analytic near the result ) are of double exponential speed at best.
You can easily see this from truncated taylors.

Infinite sums are also bounded by double exponential convergeance speed as far as i know.

I am aware ofcourse that if a sequence of fractions converges too fast , it is not algebraic.
( louiville constant , transcendental etc )

regards

tomm1729

[JmsNxn]

I know this isn't what you mean but:

\[
f(x) = \sum_{n=0}^\infty \frac{x^n}{2^{2^{2^n}}}
\]

Has triple exponential convergence!

I'm not sure we're there yet. I think with ramanujan and his numerical methods (and the progeny of ramanujan's numerical methods) are the closest we'll get. Which I think are second order factorial at best... He really changed the game for calculating \(\pi\). But also, I think this capped at double exponential--double factorial.