View source for Runge rule

One of the methods for estimating errors in numerical integration formulas (cf. [[Integration, numerical|Integration, numerical]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r0828201.png" /> be the residual term in a numerical integration formula, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r0828202.png" /> is the length of the integration interval or of some part of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r0828203.png" /> is a fixed number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r0828204.png" /> is the product of a constant with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r0828205.png" />-st derivative of the integrand at some point of the integration interval. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r0828206.png" /> is the exact value of an integral and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r0828207.png" /> is its approximate value, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r0828208.png" />.

According to Runge's rule, the same integral is calculated by the same numerical integration formula, but instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r0828209.png" /> one takes the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r08282010.png" />. Also, to obtain the value of the integral over the entire interval the integration formula is applied twice. If the derivative in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r08282011.png" /> does not change too strongly on the considered interval, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r08282012.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r08282013.png" /> is the value of the integral calculated with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082820/r08282014.png" />.

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>