Numerical integration formulas of degree two
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 formulas with Gaussian weights are \[ x^{(k)}=(x_{k,1},x_{k,2},\dots,x_{k,n}),\,k=0,1,\dots,n\tag{e17} \] with \[ x_{k,2r-1}=\sqrt{2}\cos\frac{2rk\pi}{n+1},\,\,x_{k,2r}=\sqrt{2}\sin\frac{2rk\pi}{n+1},\,r=1,2,\dots,\left[ n/2\right], \] where \(\left[ n/2\right]\) is the greatest integer not exceeding \(n/2\) (if \(n\) is odd then \(x_{k,n}=(-1)^{k}).\) A set of integration points for degree 2 formulas with beta integration weights ({e4}) are \[ y^{(k)}=\frac{1}{\alpha+\beta+2}\left[ 2\sqrt{\frac{(\alpha+1)(\beta+1)}{\alpha+\beta+3}}x^{(k)}-(\alpha-\beta)\right],\, \alpha,\beta\geq 0,\tag{e18} \] where the points \(x^{(k)}\) are defined in ({e17}). A set of integration points for degree 2 formulas with gamma integration weights ({e5}) are \[ z^{(k)}=-\sqrt{\alpha+1}\cdot x^{(k)}+(\alpha+1),\,\alpha\geq 0,\tag{e19} \] where again the points \(x^{(k)}\) are defined in ({e17}). The author lists the integration formulas of degree 3 with \(2n\) equally weighted points for integrals ({e1}) with Gaussian weights in \({\mathbb R}^{n}\) and symmetric beta weights. For Gaussian integration weights ({e3}), a set of \(2n\) equally weighted points for integration of degree 3 are \[ q^{(k)}=(q_{k,1},q_{k,2},\dots,q_{k,n}),\,\,k=1,\dots,2n,\tag{e21} \] with \( q_{k,2r-1}=\sqrt{2}\cos\frac{(2r-1)k\pi}{n},\,\,q_{k,2r}=\sqrt{2}\sin\frac{(2r-1)k\pi}{n},\,\,r=1,2,\dots,\left[ n/2\right]\) (if \(n\) is odd then \(q_{k,n}=(-1)^{k}).\) For symmetric beta integration weights ({e4}) with \(\alpha=\beta\), a set of integration points for degree 3 formulas are \(s^{(k)}=\frac{1}{\sqrt{2\alpha+3}}q^{(k)},\) where the points \(q^{(k)}\) are defined in ({e21}).
scientific article; zbMATH DE number 5349249