The Generalized Gaussian Method (GGM)

[tommy1729]

The Gaussian method can be easily generalized.

suppose we use f(s) = exp( g(s) f(s-1) ) , then we are bounded in the sense that g(s) cannot grow to fast towards 1 as Re(s) goes to +oo.
The reason is, if g(s) grows like O(exp(-exp(s)) ) then the (complex) argument (theta) gives us trouble. With erf(s) we are close to 1 + exp(-s^2) and because s^2 puts the imaginary line at 45° that is ok.

With 1 + exp(-exp(s)) however the complex argument (theta) gives us issues. 1 + exp(-exp(s)) goes to 1 fast for positive real s , BUT because of the complex argument ( theta ) this does not hold for non-real s even if their real parts are large.

So we look for functions g(s) between 1 + exp(-s^2) and 1 + exp(-exp(s)).
This is cruxial to understand !

So how do we do that ?

For starters it is also known that functions below O(exp(s)) can be completely defined by the value at 0 and its zero's.

And we want the zero's to be close to the imag axis.

This results in my generalized gaussian method.

see pictures !!

Regards

Tom Marcel Raes

tommy1729

[tommy1729]

The related integral above is quite complicated.
So I came up with the following simplification.

A different method but very similar.

n are integers larger than 0.
m is going to +infinity.

\( f(s)=e^{t(s)*f(s-1)} \)

\( t(s)=(J(s)+1)/2 \)

\( J(s) =Erf(s*p_7(s)) \)

\( p_7(s)=\prod(1+s^2/n^7) \)

\( sexp(s+s_e)=\ln^{[m]}f(s+m) \)

This has similar properties as the other generalized gaussian method and it should be easier to implement.

call it GGM2 or so.

For bases other than e ; take the base e^b then we get 

\( f_b(s)=e^{b*t(s)*f(s-1)} \)

\( t(s)=(J(s)+1)/2 \)

\( J(s) =Erf(s*p_7(s)) \)

\( p_7(s)=\prod(1+s^2/n^7) \)

\( sexp_b(s+s_b)=\ln_b^{[m]}f_b(s+m) \)

regards

tommy1729
Tom Marcel Raes

[tommy1729]

The related integral above is quite complicated.
So I came up with the following simplification.

A different method but very similar.

n are integers larger than 0.
m is going to +infinity.

\( f(s)=e^{t(s)*f(s-1)} \)

\( t(s)=(J(s)+1)/2 \)

\( J(s) =Erf(s*p_7(s)) \)

\( p_7(s)=\prod(1+s^2/n^7) \)

\( sexp(s+s_e)=\ln^{[m]}f(s+m) \)

This has similar properties as the other generalized gaussian method and it should be easier to implement.

call it GGM2 or so.

For bases other than e ; take the base e^b then we get 

\( f_b(s)=e^{b*t(s)*f(s-1)} \)

\( t(s)=(J(s)+1)/2 \)

\( J(s) =Erf(s*p_7(s)) \)

\( p_7(s)=\prod(1+s^2/n^7) \)

\( sexp_b(s+s_b)=\ln_b^{[m]}f_b(s+m) \)

regards

tommy1729
Tom Marcel Raes

A further idea is to generalize like this 
 for positive odd w ; 

\( t_w(s)=1+(J(s)-1)^w/2^w \)

for instance w = 3 or w = 7.

with w = 7 we get the case :

n are integers larger than 0.
m is going to +infinity.

\( f(s)=e^{t_w(s)*f(s-1)} \)

\( t_w(s)=1+(J(s)-1)^w/2^w \)

\( J(s) =Erf(s*p_7(s)) \)

\( p_7(s)=\prod(1+s^2/n^7) \)

\( sexp(s+s_e)=\ln^{[m]}f(s+m) \)

This has similar properties as the other generalized gaussian method and it should be easier to implement.

call it GGM2 or so.

For bases other than e ; take the base e^b then we get 

\( f_b(s)=e^{b*t_w(s)*f(s-1)} \)

\( t_w(s)=1+(J(s)-1)^w/2^w \)

\( J(s) =Erf(s*p_7(s)) \)

\( p_7(s)=\prod(1+s^2/n^7) \)

\( sexp_b(s+s_b)=\ln_b^{[m]}f_b(s+m) \)

Notice this latest new modifation does not change the range where we get close to 1 much , but is still getting faster to 1.


regards

tommy1729
Tom Marcel Raes

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