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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the numerical evaluation of a particular singular two-dimensional integral

Authors: G. Monegato and J. N. Lyness
Journal: Math. Comp. 33 (1979), 993-1002
MSC: Primary 65D30; Secondary 65B05
MathSciNet review: 528052
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Abstract: We investigate the possibility of using two-dimensional Romberg integration to approximate integrals, over the square $ 0 \leqslant x$, $ y \leqslant 1$, of integrand functions of the form $ g(x,y)/(x - y)$ where $ g(x,y)$ is, for example, analytic in x and y.

We show that Romberg integration may be properly justified so long as it is based on a diagonally symmetric rule and function values on the singular diagonal, if required, are defined in a particular way. We also investigate the consequences of ignoring fhese function values (i.e. setting them to zero) in the context of such a calculation.

We also derive the asymptotic expansion on which extrapolation methods can be based when $ g(x,y)$ has a point singularity of a specified nature at the origin.

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Article copyright: © Copyright 1979 American Mathematical Society