The distribution of points on the sphere and corresponding cubature formulae
The distribution of points on the sphere and corresponding cubature formulae
In applications, for instance in optics and astrophysics, there is a need for high-accuracy integration formulae for functions on the sphere. To construct better formulae than previously used, almost equidistantly spaced nodes on the sphere and weights belonging to these nodes are required. This problem is closely related to an optimal dispersion problem on the sphere and to the theories of spherical designs and multivariate Gauss quadrature formulae. We propose a two-stage algorithm to compute optimal point locations on the unit sphere and an appropriate algorithm to calculate the corresponding weights of the cubature formulae. Points as well as weights are computed to high accuracy. These algorithms can be extended to other integration problems. Numerical examples show that the constructed formulae yield impressively small integration errors of up to 10-12.
317-334
Fliege, Jörg
54978787-a271-4f70-8494-3c701c893d98
Maier, Ulrike
8bfa6b2f-fa5e-46ca-8e00-950df7434443
April 1999
Fliege, Jörg
54978787-a271-4f70-8494-3c701c893d98
Maier, Ulrike
8bfa6b2f-fa5e-46ca-8e00-950df7434443
Fliege, Jörg and Maier, Ulrike
(1999)
The distribution of points on the sphere and corresponding cubature formulae.
IMA Journal of Numerical Analysis, 19 (2), .
(doi:10.1093/imanum/19.2.317).
Abstract
In applications, for instance in optics and astrophysics, there is a need for high-accuracy integration formulae for functions on the sphere. To construct better formulae than previously used, almost equidistantly spaced nodes on the sphere and weights belonging to these nodes are required. This problem is closely related to an optimal dispersion problem on the sphere and to the theories of spherical designs and multivariate Gauss quadrature formulae. We propose a two-stage algorithm to compute optimal point locations on the unit sphere and an appropriate algorithm to calculate the corresponding weights of the cubature formulae. Points as well as weights are computed to high accuracy. These algorithms can be extended to other integration problems. Numerical examples show that the constructed formulae yield impressively small integration errors of up to 10-12.
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Published date: April 1999
Identifiers
Local EPrints ID: 482547
URI: http://eprints.soton.ac.uk/id/eprint/482547
ISSN: 0272-4979
PURE UUID: 073d0e88-309e-4656-b81b-e76ce3b25e16
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Date deposited: 10 Oct 2023 16:52
Last modified: 06 Jun 2024 01:46
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Author:
Ulrike Maier
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