Numerical Integration on the Sphere

  • Kerstin HesseEmail author
  • Ian H. Sloan
  • Robert S. Womersley
Reference work entry


This chapter is concerned with numerical integration over the unit sphere \(\mathbb{S}^{2} \subset \mathbb{R}^{3}\). We first discuss basic facts about numerical integration rules with positive weights. Then some important types of rules are discussed in detail: rules with a specified polynomial degree of precision, including the important case of longitude-latitude rules; rules using scattered data points; rules based on equal-area partitions; and rules for numerical integration over subsets of the sphere. Finally we show that for numerical integration over the whole sphere and for functions with an appropriate degree of smoothness, an optimal rate of convergence can be achieved by positive-weight rules with polynomial precision and also by rules obtained by integrating a suitable radial basis function interpolant.



The support of the Australian Research Council is gratefully acknowledged. IHS and RSW acknowledge the support of the Hong Kong Polytechnic University, where much of this work was carried out. The authors also thank Ronald Cools for helpful advice.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Kerstin Hesse
    • 1
    Email author
  • Ian H. Sloan
    • 2
  • Robert S. Womersley
    • 3
  1. 1.Department of MathematicsUniversity of PaderbornPaderbornGermany
  2. 2.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  3. 3.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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