Geodesic
Quadrature over the sphere
Electronic transactions on numerical analysis, Tome 20 (2005), pp. 104-118
Consider integration over the unit sphere in , especially when the integrand has singular behaviour $\sterling $######## 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 §$\copyright \ddot $###

$ 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.$

Classification : 65D32
Keywords: spherical integration, trapezoidal rule, Euler-MacLaurin expansion 
@article{ETNA_2005__20__a9,
     author = {Atkinson,  Kendall and Sommariva,  Alvise},
     title = {Quadrature over the sphere},
     journal = {Electronic transactions on numerical analysis},
     pages = {104--118},
     year = {2005},
     volume = {20},
     zbl = {1078.65018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2005__20__a9/}
}
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%A Sommariva,  Alvise
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Atkinson,  Kendall; Sommariva,  Alvise. Quadrature over the sphere. Electronic transactions on numerical analysis, Tome 20 (2005), pp. 104-118. http://geodesic.mathdoc.fr/item/ETNA_2005__20__a9/