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+--- Thread: numerical methods with triple exp convergeance ? (/showthread.php?tid=1731)
numerical methods with triple exp convergeance ? - tommy1729 - 03/25/2023
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
RE: numerical methods with triple exp convergeance ? - JmsNxn - 03/27/2023
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.