Numerical integration formulas of degree two
type: Item
instance of: scholarly article
title: Numerical integration formulas of degree two
zbMATH Open document ID: 1221.65082
author: Dongbin Xiu
published in: Applied Numerical Mathematics
publication date: 2008-10-07T00:00:00Z
review text: Let \[ \int_{D}f(x)W(x)dx\approx\sum_{k}a_{k}f(u^{(k)})\tag{e1} \] be a numerical integration formula, where \(D\subset{\mathbb R}^{n}\) is a region in an \(n\)-dimensional, real, Euclidian space, \(x=(x_{1},x_{2},\dots,x_{n})\) denotes a coordinate, \(a_{k}\) are constants, and \(u^{(k)}\) are points in the space. The formulas are said to be of degree \(N\) if they are exact for integrations of any polynomial in \(x\) of degree at most \(N\) but not \(N+1\). For \(n\)-dimensional symmetric integrals, Stroud analyzed the disposition of the points and gave sets of points for degree 2 and 3 formulas consisting of \(n+1\) and \(2n\) equally weighted points, respectively. In this paper, the author extends Stroud's results and presents formulas of degree 2 of \(n+1\) equally weighted points for integrals with nonsymmetric integration weights. In particular, the author considers integrals with non-negative weights of identical and independent components, i.e., \[ W(x)=w(x_{1})\cdots w(x_{n}),\, w(x_{i})\geq 0,\, i=1,\dots,n. \] Such integrals often arise in statistical analysis where one is required to evaluate expectations of multivariate distributions, where \(W(x)\) represents a probability density function with identical and independent components. The specific cases considered in this article include, for \(i=1,\dots,n\), \(\bullet\) Gaussian weights in \({\mathbb R}^{n}\) \[ w(x_{i})=\frac{1}{\sqrt{2\pi}}exp\left(-\frac{x_{i}^{2}}{2}\right),\,-\infty<x_{i}<\infty\tag{e3} \] \(\bullet\) beta weights in \(n\)-cube \[ w(x_{i})\sim (1-x_{i})^{\alpha}(1+x_{i})^{\beta},\,\alpha,\beta\geq 0,\, x_{i}\in[-1,1]\tag{e4} \] \(\bullet\) gamma weights in \([0,\infty)^{n}\) \[ w(x_{i})\sim e^{-x_{i}}x_{i}^{\alpha},\,\alpha\geq 0,\, x_{i}\geq 0.\tag{e5} \] In this paper, the author presents explicit points locations for formulas of degree 2 with \(n+1\) equally weighted points for the Gaussian, beta and gamma integration weights. A set of integration points for the degree 2 formula…
reviewed by: Ana Maria Acu
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 5349249
zbMATH Keywords: numerical integration
zbMATH Keywords: cubature formulas
MaRDI profile type: Publication
full work available at URL: j.apnum.2007.09.004
OpenAlex ID: W2166033439
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DOI: 10.1016/J.APNUM.2007.09.004
description: scientific article; zbMATH DE number 5349249