Namespaces
Variants
Actions

Difference between revisions of "Runge rule"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
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" />.
+
{{TEX|done}}
 +
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 <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
+
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
  
<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>
+
$$R=h^kM=\frac{I_1-I}{1-\frac{1}{2^{k-1}}},$$
  
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" />.
+
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).
 
Runge's rule is also used when numerically solving differential equations. The rule was proposed by C. Runge (beginning of the 20th century).

Latest revision as of 13:59, 27 August 2014

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)
How to Cite This Entry:
Runge rule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Runge_rule&oldid=33167
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article