Summary: Electronic Transactions on Numerical Analysis.
Volume 20, pp. 104-118, 2005.
Copyright © 2005, Kent State University.
Kent State University
QUADRATURE OVER THE SPHERE
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 , 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  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 , although the latter is for a different, but related, class of transformations.