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 , , of integrand functions of the form where 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 has a point singularity of a specified nature at the origin.

**[1]**R. L. BISPLINGHOFF, H. ASHLEY & R. L. HALFMAN,*Aeroelasticity*, Addison-Wesley, Reading, Mass., 1957, pp. 188-293.**[2]**J. N. Lyness,*Symmetric integration rules for hypercubes. III. Construction of integration rules using null rules*, Math. Comp.**19**(1965), 625–637. MR**0201069**, 10.1090/S0025-5718-1965-0201069-7**[3]**J. N. Lyness,*An error functional expansion for 𝑁-dimensional quadrature with an integrand function singular at a point*, Math. Comp.**30**(1976), no. 133, 1–23. MR**0408211**, 10.1090/S0025-5718-1976-0408211-0**[4]**J. N. Lyness and B. W. Ninham,*Numerical quadrature and asymptotic expansions*, Math. Comp.**21**(1967), 162–178. MR**0225488**, 10.1090/S0025-5718-1967-0225488-X**[5]**Charles C. S. Song,*Numerical integration of a double integral with Cauchy-type singularity*, AIAA J.**7**(1969), 1389–1390. MR**0245204****[6]**William Squire,*An efficient iterative method for numerical evaluation of integrals over a semi-infinite range*, Internat. J. Numer. Methods Engrg.**10**(1976), no. 2, 478–484. MR**0455307**

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1979-0528052-6

Article copyright:
© Copyright 1979
American Mathematical Society