Abstract
We investigate the possibility of using two-dimensional Romberg integration to approximate integrals, over the square 0 ⊽ x 0 \leqslant x , y ⊽ 1 y \leqslant 1 , of integrand functions of the form g ( x , y ) / ( x â y ) g(x,y)/(x - y) where g ( x , y ) 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 ) g(x,y) has a point singularity of a specified nature at the origin.
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