Runge rule

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

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

where is the value of the integral calculated with respect to .

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)