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Summary: Electronic Transactions on Numerical Analysis.
Volume 20, pp. 104-118, 2005.
Copyright © 2005, Kent State University.
ISSN 1068-9613.
ETNA
Kent State University
etna@mcs.kent.edu
QUADRATURE OVER THE SPHERE
KENDALL ATKINSON
¡
AND ALVISE SOMMARIVA¢
Abstract. Consider integration over the unit sphere in £¥¤ , especially when the integrand has singular behaviour
in a polar region. In an earlier paper [4], a numerical integration method was proposed that uses a transformation
that leads to an integration problem over the unit sphere with an integrand that is much smoother in the polar regions
of the sphere. The transformation uses a grading parameter ¦ . The trapezoidal rule is applied to the spherical
coordinates representation of the transformed problem. The method is simple to apply, and it was shown in [4] to
have convergence §©¨ or better for integer values of ¦ . In this paper, we extend those results to non-integral
values of ¦ . We also examine superconvergence that was observed when ¦ is an odd integer. The overall results
agree with those of [11], although the latter is for a different, but related, class of transformations.
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