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![The real (left) and the imaginary (right) parts of the actual error for the Gaussian rule (full line) and the estimates of these errors obtained by application of the corresponding anti-Gaussian rules on the semicircle (dashed line) in calculation of I(f) for f(z)=iz/(z-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)={\textrm{i} z}/{(z-2)}$$\end{document} and w(z)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(z)=1$$\end{document} for n=4(2)10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=4(2)10$$\end{document}](https://www.researchgate.net/publication/384864731/figure/fig5/AS:11431281283384267@1728784704315/The-real-left-and-the-imaginary-right-parts-of-the-actual-error-for-the-Gaussian-rule_Q320.jpg)
October 2024
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22 Reads
Numerical Algorithms
Let D+ be defined as D+={z∈C:|z|<1,Imz>0} and Γ be a unit semicircle Γ={z=eiθ:0≤θ≤π}=∂D+. Let w(z) be a weight function which is positive and integrable on the open interval (-1,1), though possibly singularity at the endpoints, and which can be extended to a function w(z) holomorphic in the half disc D+. Orthogonal polynomials on the semicircle with respect to the complex-valued inner product (f,g)=∫Γf(z)g(z)w(z)(iz)-1dz=∫0πf(eiθ)g(eiθ)w(eiθ)dθwas introduced by Gautschi and Milovanović in ( J. Approx. Theory 46, 230-250, 1986) (for w(z)=1w(z)=1), where the certain basic properties were proved. Such orthogonality as well as the applications involving Gauss-Christoffel quadrature rules were further studied in Gautschi et al. (Constr. Approx. 3, 389-404, 1987) and Milovanović (2019). Inspired with Laurie’s paper (Math. Comp. 65(214), 739-747, 1996), this article introduces anti-Gaussian quadrature rules related to orthogonality on the semicircle, presents some of their properties, and suggests a numerical method for their construction. We demonstrate how these rules can be used to estimate the error of the corresponding Gaussian quadrature rules on the semicircle. Additionally, we introduce averaged Gaussian rules related to orthogonality on the semicircle to reduce the error of the corresponding Gaussian rules. Several numerical examples are provided.