View source for Runge rule

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One of the methods for estimating errors in numerical integration formulas (cf. [[Integration, numerical|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.S. Berezin,   N.P. Zhidkov,   "Computing methods" , Pergamon  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Hall (ed.)  J.M. Watt (ed.) , ''Modern numerical methods for ordinary differential equations'' , Clarendon Press  (1976)</TD></TR></table>