It turns out the correcting factor is exactly sqrt( 2 pi n ) just like for exp.
So the gaussian is a powerfull idea.
In the past I expressed doubt due to the existance of functions with a more complicated Riemann surface. In particular because of the contour.
Neither that doubt nor the dismissal has been considered enough , hence they need more study.
No lack of work to do in fake function theory.
Im not An expert in deepest descent methods but this might be intresting here.
I have many more ideas but I am most confident in this one :
Tommy-Sheldon iterations
( first order )
---
The dot product ( • ) for Taylor series :
Z(x) = z_0 + z_1 x + ...
Z(x) • u(n) = z_0 u_0 + z_1 u_1 x + ...
---
Given our valid function f(x) of wich we want a fake
F_0(x) = f(x)
G_0(x) = ln( f(exp(x)) )
G_k(x) = ln( F_k(exp(x)) )
Now use the min(f / x^n) method from post 9 = S9.
No rescaling.
F_1(x) = S9(F_0(x))
F_2(x) = F_1(x) • [ sqrt( 2 pi G_1 '' (h_n) ) ]^ -1
F_3(x) = F_1(x) • [ sqrt( 2 pi G_2 '' (h_n) ) ]^ -1
F_4(x) = F_1(x) • [ sqrt( 2 pi G_3 '' (h_n) ) ]^ -1
...
F_oo(x) = ts( f(x) ) = tsf(0) + tsf(1) x + ...
I believe if €f(x) = the best fake for f with coëfficiënts
€(n) then ts1(n)/€(n) =< (1 + O(1/n)).
As for higher orders those are LIKELY both convergeance accelerators of F_n(x) AND
Give higher precision [ 1 + O(1/n^2) i guess ] , also probably by adding higher derivatives.
Notice the Tommy-Sheldon iterations do not require f ''' (y) > 0 for all y > 0.
I assumed it is not possible to increase convergence speed without precision or complexity ( higher derivatives ).
This recursion reminds me of numerical methods used for differential equations.
Its weird , how this nonstandard idea connects to classical ideas.
But I guess we are used to that on the tetration forum.
As far as I know this is the best method.
Sheldon latest methods IV,V,... Are I assume only good/valid for tetration type functions , not for general functions.
Notice the Tommy-Sheldon iterations solve the issue of " too small f '' (h_n) ".
I hope you guys do not Mind me ignoring this Roman numerals hype here.
Regards
Tommy1729
So the gaussian is a powerfull idea.
In the past I expressed doubt due to the existance of functions with a more complicated Riemann surface. In particular because of the contour.
Neither that doubt nor the dismissal has been considered enough , hence they need more study.
No lack of work to do in fake function theory.
Im not An expert in deepest descent methods but this might be intresting here.
I have many more ideas but I am most confident in this one :
Tommy-Sheldon iterations
( first order )
---
The dot product ( • ) for Taylor series :
Z(x) = z_0 + z_1 x + ...
Z(x) • u(n) = z_0 u_0 + z_1 u_1 x + ...
---
Given our valid function f(x) of wich we want a fake
F_0(x) = f(x)
G_0(x) = ln( f(exp(x)) )
G_k(x) = ln( F_k(exp(x)) )
Now use the min(f / x^n) method from post 9 = S9.
No rescaling.
F_1(x) = S9(F_0(x))
F_2(x) = F_1(x) • [ sqrt( 2 pi G_1 '' (h_n) ) ]^ -1
F_3(x) = F_1(x) • [ sqrt( 2 pi G_2 '' (h_n) ) ]^ -1
F_4(x) = F_1(x) • [ sqrt( 2 pi G_3 '' (h_n) ) ]^ -1
...
F_oo(x) = ts( f(x) ) = tsf(0) + tsf(1) x + ...
I believe if €f(x) = the best fake for f with coëfficiënts
€(n) then ts1(n)/€(n) =< (1 + O(1/n)).
As for higher orders those are LIKELY both convergeance accelerators of F_n(x) AND
Give higher precision [ 1 + O(1/n^2) i guess ] , also probably by adding higher derivatives.
Notice the Tommy-Sheldon iterations do not require f ''' (y) > 0 for all y > 0.
I assumed it is not possible to increase convergence speed without precision or complexity ( higher derivatives ).
This recursion reminds me of numerical methods used for differential equations.
Its weird , how this nonstandard idea connects to classical ideas.
But I guess we are used to that on the tetration forum.
As far as I know this is the best method.
Sheldon latest methods IV,V,... Are I assume only good/valid for tetration type functions , not for general functions.
Notice the Tommy-Sheldon iterations solve the issue of " too small f '' (h_n) ".
I hope you guys do not Mind me ignoring this Roman numerals hype here.
Regards
Tommy1729
