Runge rule

One of the methods for estimating errors in numerical integration formulas (cf. Integration, numerical). Let $R=h^kM$ be the residual term in a numerical integration formula, where $h$ is the length of the integration interval or of some part of it, $k$ is a fixed number and $M$ is the product of a constant with the $(k-1)$-st derivative of the integrand at some point of the integration interval. If $J$ is the exact value of an integral and $I$ is its approximate value, then $J=I+h^kM$.

According to Runge's rule, the same integral is calculated by the same numerical integration formula, but instead of $h$ one takes the value $h/2$. Also, to obtain the value of the integral over the entire interval the integration formula is applied twice. If the derivative in $M$ does not change too strongly on the considered interval, then

$$R=h^kM=\frac{I_1-I}{1-\frac{1}{2^{k-1}}},$$

where $I_1$ is the value of the integral calculated with respect to $h/2$.

Runge's rule is also used when numerically solving differential equations. The rule was proposed by C. Runge (beginning of the 20th century).

References

[1]	I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)
[2]	G. Hall (ed.) J.M. Watt (ed.) , Modern numerical methods for ordinary differential equations , Clarendon Press (1976)