Cubature formulae for the Gaussian weight. Some old and new rules.

Electronic Transactions on Numerical Analysis

article

Ramón Orive¹  Juan C. Santos-León¹  Miodrag M. SPALEVIC² 

[1] Departmento de Análisis Matemático, Universidad de La Laguna;Department of Mathematics, Faculty of Mechanical Engineering, University of Beograd

关键词: cubature formulas;    Gaussian weight;   

DOI  :  10.1553/etna_vol53s426

学科分类：数学（综合）

来源: Kent State University * Institute of Computational Mathematics

【 摘 要 】

In this paper we review some of the main known facts about cubature rules to approximate integrals over domains in $\mathbb{R}^n$, in particular with respect to the Gaussian weight $\displaystyle w(\mathbf{x}) = e^{-\mathbf{x}^T\,\mathbf{x}},$ where $\mathbf{x} = (x_1,\ldots,x_n)\in \mathbb{R}^n.$ Some new rules are also presented. Taking into account the well-known issue of the “curse of dimensionality”, our aim is at providing rules with a certain degree of algebraic precision and a reasonably small number of nodes as well as an acceptable stability. We think that the methods used to construct these new rules are of further applicability in the field of cubature formulas. The efficiency of new and old rules are compared by means of several numerical experiments.

【 授权许可】

Unknown