Ékstrapolasi Richardson

Dina analisis numeris, ékstrapolasi Richardson nyaéta métode akselerasi runtuyan, nu digunakeun pikeun ngabebenah rarata konvergénsi tina runtuyan. Dingaranan sanggeus kapanggih ku Lewis Fry Richardson, nu manggihan téhnik ieu dina mangsa awal abad ka-20.

 Définisi basajan 

Suppose that A(h) is an estimation of order hn for
, i.e. . Then

is called the Richardson extrapolate of A(h);
it is an estimate of order  hm for A, with m>n.

More generally, the factor 2 can be replaced by any other factor, as shown below.

Very often, it is much éasier to obtain a given precision by using R(h) rather
than A(h') with a much smaller  h' , which can cause problems due to limited precision (rounding errors) and/or due to the incréasing number of calculations needed (see examples below).

 Rumus umum 
Let A(h) be an approximation of A that depends on a positive step size h with an error formula of the form 

where the ai are unknown constants and the ki are known constants such that hki > hki+1.

The exact value sought can be given by

which can be simplified with Big O notation to be

Using the step sizes h and h / t for some t, the two formulas for A are:

Multiplying the second equation by t<sup>k<sub>0</sub></sup> and subtracting the first equation gives

which can be solved for A to give

By this process, we have achieved a better approximation of A by subtracting the largest term in the error which was O(hk0).  This process can be repéated to remove more error terms to get even better approximations.

A general recurrence relation can be defined for the approximations by

such that 

A well-known practical use of Richardson extrapolation is Romberg integration, which applies Richardson extrapolation to the trapezium rule.

It should be noted that the Richardson extrapolation can be considered as a linéar sequence transformation.

 Conto 

Using Taylor's theorem,

so the derivative of f(x) is given by

If the initial approximations of the derivative are chosen to be

then ki = i+1.

For t = 2, the first formula extrapolated for A would be

For the new approximation 

we can extrapolate again to obtain

 Rujukan 

Extrapolation Methods. Theory and Practice ku C. Brezinski jeung M. Redivo Zaglia, North-Holland, 1991.

 Tempo ogé 
 Aitken's delta-squared process
 Nyungsi 'Richardson extrapolation' di google

Kategori:Analisis numeris
Kategori:Analisis asimtotis