Extrapolation of Richardson

In numerical Analysis, the proceeded of extrapolation of Richardson is a technique of acceleration of convergence. He is thus named in the honor of Lewis Fry Richardson, who introduced it at the beginning of the XXe century.

This process is in particular used to define a numerical method of integration: the Method of Romberg, acceleration of the Method of the trapezoids.

Presentation of the principle

One supposes that the unknown quantity has can be approximate by a function has (H) with a convergence of order N in H

$A-A\; (H)\; =\; a\_n\; h^n+O\; (h^m),\; ~a\_n\; \backslash \; ne0,\; ~m>n,$

expression in which the coefficient a_n is not known. The principle of extrapolation consists in forming

$R\; (H)\; =\; has\; (h/2)\; +\; \backslash \; frac\; \{has\; (h/2)\; -\; has\; (H)\}\{2^n-1\}\; =\; \backslash \; frac\; \{2^n\; \backslash ,\; has\; (h/2)\; -\; has\; (H)\}\{2^n-1\}$

who approaches has with the order m>n in H .

General formula and iteration

One supposes that one has an approximation of has with a formula of error of this form

$has\; =\; has\; (H)\; +\; a\_0h^\; \{k\_0\}\; +\; a\_1h^\; \{k\_1\}\; +\; a\_2h^\; \{k\_2\}\; +\; \backslash \; cdots\; a\_zh^\; \{k\_z\}\; +O\; (h^\; \{k\_\; \{z+1\}\}),$

coefficients being unknown. One sets a real parameter r>1 and one forms a combination between the preceding relation and this same relation taken at the point $h/r$

$(r^\; \{k\_0\}\; -\; 1)\; has\; =\; r^\; \{k\_0\}\; has\; \backslash \; left\; (\backslash \; frac\; \{H\}\; \{R\}\; \backslash \; right)\; -\; has\; (H)\; +\; 0+a\_1\; \backslash \; left\; (\backslash \; frac\; \{r^\; \{k\_0\}\}\; \{r^\; \{k\_1\}\}\; -\; 1\; \backslash \; right)\; h^\; \{k\_1\}\; +\; \backslash \; dots+a\_z\; \backslash \; left\; (\backslash \; frac\; \{r^\; \{k\_0\}\}\; \{r^\; \{k\_z\}\}\; -\; 1\; \backslash \; right)\; h^\; \{k\_z\}\; +O\; (h^\; \{k\_\; \{z+1\}\}).$

This formula can be reiterated to increase the order.

See too

Related articles

• Delta-2

• Method of Romberg

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