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List:       kde-devel
Subject:    [OT] Richardson extrapolation
From:       Stefano Borini <munehiro () ferrara ! linux ! it>
Date:       2003-09-01 8:05:13
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On Mon, Sep 01, 2003 at 12:40:45AM +0200, Stefano Borini wrote:
> > and use b for the value of I for that step (which corresponds n going to 
> > infinity).  But, a more rigorous explanation would be nice.

here it is.
suppose we have a value that depends on a parameter d through the asymptotic
law

E(d) = E_0 + E_1 d^2 + E_2 d^4 + ...

Where d goes to zero as we improve accuracy. The obtainment of E_0 is affected
by an error in the order of o(d^2)

at 2d we have

E(2d) = E_0 + 4 * E_1 d^2 + 16 * E_2 d^4 + ...

multiplying the first equation by 4 and subtracting the second yelds:

(4 * E(d) - E(2d) ) / 3 = E_0 + o(d^4)

of course you can choose a generic multiplying factor of alpha, instead of 2,
and it's trivial to get the equation i gave in the previous mail.

 
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