Advances in Computational Mathematics

, Volume 36, Issue 3, pp 451–483 | Cite as

Numerical integration with polynomial exactness over a spherical cap

  • Kerstin HesseEmail author
  • Robert S. Womersley


This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \(\mathbb{S}^2\), we discuss tensor product rules with n 2/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n 3) nodes for numerical integration over spherical caps on \(\mathbb{S}^2\). For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \(\mathbb{S}^d\) that have O(n d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \(\mathbb{S}^d\) that are exact for all spherical polynomials of degree ≤ n have at least O(n d ) nodes and possess a certain regularity property.


Equal weight rules Markov inequalities Numerical integration Polynomial exactness Positive weights Spherical cap Tensor product rules 

Mathematics Subject Classifications (2010)

Primary 65D30 65D32 33C55; Secondary 41A55 33C50 42C10 43A90 58C35 


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© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Handelshochschule Leipzig gGmbHLeipzigGermany
  2. 2.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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