(10/26/2021, 10:41 PM)tommy1729 Wrote: 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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