Tommy's Gaussian method.

[tommy1729]

An interesting idea is this :

Are all these " beta methods " equivalent ? as James asks.

And 

Is there a way to accelerate the convergeance of the iterations ? 

Series acceleration is well known but iteration acceleration not so much.

Those are 2 nice questions , but what is the interesting idea you might ask ??

Well that those 2 questions are related !!

let 

f_1(s) = exp( t(1 s) * f(s-1))

f_2(s) = exp( t(2 s) * f(s-1))

and the resp analytic tetrations from them : F1(s) and F2(s).

Remember that tetration(s + theta(s)) is also tetration where theta(s) is a suitable analytic real 1-periodic function. 

so F2(s) = F1(s + theta1(s))  , F1(s) = F2(s + theta2(s)).

BUT THIS ALSO IMPLIES THAT 

f*_2(s) = exp( t( 2*(s + theta2(s)) ) * f(s-1))    .. RESULTING IN F2_*(s) is actually equal to 

F2_*(s) = F2(s + theta(s)) = F1(s)

HENCE USING t(1 s) = t(s) is the same as using t( 2 s + 2 theta(s)) !!

So this relates to the main questions posed :

when are 2 solutions equal ?

How to accelerate convergeance ?

As for the acceleration , ofcourse the complexity and difficulty of theta and computing theta are key.

But numerically it is expected using t( 2 s + 2 theta(s) ) converges faster. ( because using t(2s) does converge faster than using t(s) ) .

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Tom(s,v) = exp( t(v * s)   *   exp(Tom(s-1,v)) )

resulting in 

tet(s+1,v) = exp( tet(s,v) ).

I Like that notation.

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regards

tommy1729