Ékstrapolasi Richardson

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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[édit | sunting sumber]

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Bantosanna diantos kanggo narjamahkeun.

Suppose that A(h) is an estimation of order hn for A=\lim_{h\to 0}A(h), i.e. A-A(h) = a_n h^n+O(h^m),~a_n\ne0,~m>n. Then

R(h) = A(h/2) + \frac{A(h/2)-A(h)}{2^n-1} = \frac{2^n\,A(h/2)-A(h)}{2^n-1}

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 easier 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 increasing number of calculations needed (see examples below).

Rumus umum[édit | sunting sumber]

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

 A - A(h) = a_0h^{k_0} + a_1h^{k_1} + a_2h^{k_2} + \cdots

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

 A = A(h) + a_0h^{k_0} + a_1h^{k_1} + a_2h^{k_2} + \cdots

which can be simplified with Big O notation to be

 A = A(h)+ a_0h^{k_0} + O(h^{k_1}).  \,\!

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

 A = A(h)+ a_0h^{k_0} + O(h^{k_1})  \,\!
 A = A\left(\frac{h}{t}\right) + a_0\left(\frac{h}{t}\right)^{k_0} + O(h^{k_1}) .

Multiplying the second equation by tk0 and subtracting the first equation gives

 (t^{k_0}-1)A = t^{k_0}A\left(\frac{h}{t}\right) - A(h) + O(h^{k_1})

which can be solved for A to give

A = \frac{t^{k_0}A\left(\frac{h}{t}\right) - A(h)}{t^{k_0}-1} + O(h^{k_1}) .

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 repeated to remove more error terms to get even better approximations.

A general recurrence relation can be defined for the approximations by

 A_{i+1}(h) = \frac{t^{k_i}A_i\left(\frac{h}{t}\right) - A_i(h)}{t^{k_i}-1}

such that

 A = A_{i+1}(h) + O(h^{k_{i+1}}) .

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 linear sequence transformation.

Conto[édit | sunting sumber]

Using Taylor's theorem,

f(x+h) = f(x) + f'(x)h + \frac{f''(x)}{2}h^2 + \cdots

so the derivative of f(x) is given by

f'(x) = \frac{f(x+h) - f(x)}{h} - \frac{f''(x)}{2}h + \cdots.

If the initial approximations of the derivative are chosen to be

A_0(h) = \frac{f(x+h) - f(x)}{h}

then ki = i+1.

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

A = 2A_0\left(\frac{h}{2}\right) - A_0(h) + O(h^2) .

For the new approximation

A_1(h) = 2A_0\left(\frac{h}{2}\right) - A_0(h)

we can extrapolate again to obtain

 A = \frac{4A_1\left(\frac{h}{2}\right) - A_1(h)}{3} + O(h^3) .

Rujukan[édit | sunting sumber]

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

Tempo ogé[édit | sunting sumber]