ArticlePublisher preview available

Anti-Gaussian quadrature rules related to orthogonality on the semicircle

Authors:
Aleksandra S. Milosavljević
Aleksandra S. Milosavljević
Aleksandra S. Milosavljević
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Marija P. Stanić
Marija P. Stanić
Marija P. Stanić
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Tatjana V. Tomovic Mladenovic at University of Kragujevac
Tatjana V. Tomovic Mladenovic
Read publisher preview
Download citation
To read the full-text of this research, you can request a copy directly from the authors.

Abstract and Figures

Let D+D+D_+ be defined as D+={z∈C:|z|<1,Imz>0}D+={z∈C : ∣z∣<1,Im z>0}D_+=\{z\in \mathbb {C}\,:\,|z|<1,\textrm{Im}\,z>0\} and ΓΓ\Gamma be a unit semicircle Γ={z=eiθ:0≤θ≤π}=∂D+Γ={z=eiθ:0≤θ≤π}=∂D+\Gamma =\{z={\textrm{e}}^{{\textrm{i}}\theta }: 0\le \theta \le \pi \}=\partial D_+. Let w(z) be a weight function which is positive and integrable on the open interval (-1,1)(−1,1)(-1,1), though possibly singularity at the endpoints, and which can be extended to a function w(z) holomorphic in the half disc D+D+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)dθ(f,g)=∫Γf(z)g(z)w(z)(iz)−1dz=∫0πf(eiθ)g(eiθ)w(eiθ)dθ\begin{aligned} ( f,g)=\int _{\Gamma } f(z) g(z)w(z)(\textrm{i} z)^{-1}\textrm{d} z=\int \limits _0^{\pi } f(\textrm{e} ^{\textrm{i} \theta })g(\textrm{e} ^{\textrm{i} \theta })w(\textrm{e} ^{\textrm{i} \theta })\textrm{d} \theta \end{aligned}was 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.
The contour Cε=[-1+ε,1-ε]∪cε,1∪Γε∪cε,-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\varepsilon }=[-1+\varepsilon ,1-\varepsilon ] \cup c_{\varepsilon ,1} \cup \Gamma _{\varepsilon } \cup c_{\varepsilon ,-1}$$\end{document}
The contour Cε=[-1+ε,1-ε]∪cε,1∪Γε∪cε,-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\varepsilon }=[-1+\varepsilon ,1-\varepsilon ] \cup c_{\varepsilon ,1} \cup \Gamma _{\varepsilon } \cup c_{\varepsilon ,-1}$$\end{document}
… 
The real (left) and the imaginary (right) parts of the actual errors for the Gaussian (red), the anti-Gaussian (blue) and the averaged Gaussian rules (green) related to orthogonality on the semicircle with respect to the weight function w(z)=(1-z2)1.3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(z)=(1-z^2)^{1.3}$$\end{document} for f(z)=(z+5)/(z+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=(z+5)/(z+1)$$\end{document}, n=20(5)100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=20(5)100$$\end{document}
The real (left) and the imaginary (right) parts of the actual errors for the Gaussian (red), the anti-Gaussian (blue) and the averaged Gaussian rules (green) related to orthogonality on the semicircle with respect to the weight function w(z)=(1-z2)1.3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(z)=(1-z^2)^{1.3}$$\end{document} for f(z)=(z+5)/(z+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=(z+5)/(z+1)$$\end{document}, n=20(5)100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=20(5)100$$\end{document}
… 
Real (left) and imaginary (right) parts of the actual errors for Gaussian rules (full line) and the estimates of these errors (dashed line) in calculation of I0(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_0(1)$$\end{document} for n=2(1)5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2(1)5$$\end{document}
Real (left) and imaginary (right) parts of the actual errors for Gaussian rules (full line) and the estimates of these errors (dashed line) in calculation of I0(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_0(1)$$\end{document} for n=2(1)5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2(1)5$$\end{document}
… 
The real parts of the actual errors for the Gaussian rules (blue) and the estimates of these errors obtained by application of the corresponding anti-Gaussian rules on the semicircle (orange) for w(z)=(1-z2)-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(z)=(1-z^2)^{-{1}/{2}}$$\end{document}, in calculation of I0(0.25;f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_0(0.25;f)$$\end{document} for f(z)=1/(z2+a2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)={1}/{(z^2+a^2)}$$\end{document}, a=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=5$$\end{document} for n=2(1)8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2(1)8$$\end{document}
The real parts of the actual errors for the Gaussian rules (blue) and the estimates of these errors obtained by application of the corresponding anti-Gaussian rules on the semicircle (orange) for w(z)=(1-z2)-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(z)=(1-z^2)^{-{1}/{2}}$$\end{document}, in calculation of I0(0.25;f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_0(0.25;f)$$\end{document} for f(z)=1/(z2+a2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)={1}/{(z^2+a^2)}$$\end{document}, a=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=5$$\end{document} for n=2(1)8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2(1)8$$\end{document}
… 
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}
+3
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}
… 
Figures - available from: Numerical Algorithms
This content is subject to copyright. Terms and conditions apply.
A preview of this full-text is provided by Springer Nature.
Learn more
Content available from Numerical Algorithms
This content is subject to copyright. Terms and conditions apply.
Numerical Algorithms
https://doi.org/10.1007/s11075-024-01948-z
RESEARCH
Anti-Gaussian quadrature rules related to orthogonality
on the semicircle
Aleksandra S. Milosavljevi´c1·Marija P. Stani´c1·
Tatjana V. Tomovi´c Mladenovi´c1
Received: 23 April 2024 / Accepted: 24 September 2024
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature
2024
Abstract
Let D+be defined as D+={z∈C:|z|<1,Im z>0}and be a unit semicircle
={z=eiθ:0≤θ≤π}=∂D+.Letw(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)=
f(z)g(z)w(z)(iz)−1dz=
π
0
f(eiθ)g(eiθ)w(eiθ)dθ
was introduced by Gautschi and Milovanovi´c in ( J. Approx. Theory 46, 230-250,
1986) (for w(z)=1), where the certain basic properties were proved. Such orthog-
onality as well as the applications involving Gauss-Christoffel quadrature rules were
further studied in Gautschi et al. (Constr. Approx. 3, 389-404, 1987) and Milovanovi´c
(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 semi-
circle, 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.
1 Introduction
Let D+be defined as D+={z∈C:|z|<1,Im z>0}and be a unit semicircle
={z=eiθ:0≤θ≤π}=∂D+. Orthogonal polynomials on the semicircle with
The authors were supported in part by the Serbian Ministry of Science, Technological Development and
Innovation, contract number 451-03-65/2024-03/200122.
Extended author information available on the last page of the article
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
ResearchGate has not been able to resolve any citations for this publication.
Quantum signal processing and nonlinear Fourier analysis
Article
Full-text available
  • Jun 2024
Elucidating a connection with nonlinear Fourier analysis (NLFA), we extend a well known algorithm in quantum signal processing (QSP) to represent measurable signals by square summable sequences. Each coefficient of the sequence is Lipschitz continuous as a function of the signal.
View
Orthogonality on the semicircle: old and new results
Article
Full-text available
  • Jan 2023
  • ELECTRON T NUMER ANA
Orthogonal polynomials on the semicircle were introduced by Gautschi and Milovanovic ́ in [Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (July 1985), pp. 179-185] and [J. Approx. Theory, 46 (1986), pp. 230-250]. In this paper we give an account of this kind of orthogonality, weighted generalizations mainly oriented to Chebyshev weights of the first and second kinds, including several interesting properties of such polynomials. Moreover, we also present a number of new results including those for Laurent polynomials (rational functions) orthogonal on the semicircle. In particular, we give their recurrence relations and study special cases for the Legendre weight and for the Chebyshev weights of the first and second kind. Explicit expressions for such orthogonal systems with Chebyshev weights are presented, as well as the corresponding zero distributions.
View
Anti-Gaussian quadrature rule for trigonometric polynomials
Article
Full-text available
  • Jan 2022
In this paper, anti-Gaussian quadrature rules for trigonometric polynomials are introduced. Special attention is paid to an even weight function on [-?, ?). The main properties of such quadrature rules are proved and a numerical method for their construction is presented. That method is based on relations between nodes and weights of the quadrature rule for trigonometric polynomials and the quadrature rule for algebraic polynomials. Some numerical examples are included. Also, we compare our method with other available methods.
View
The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization
Article
Full-text available
  • Jan 2021
  • NUMER ALGORITHMS
The concept of Gauss quadrature can be generalized to approximate linear functionals with complex moments. Following the existing literature, this survey will revisit such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian) Lanczos algorithm. Analogously, the Gauss quadrature for linear functionals is connected with formal orthogonal polynomials, and with the non-Hermitian Lanczos algorithm with look-ahead strategy; moreover, it is related to the minimal partial realization problem. We will review these connections pointing out the relationships between several results established independently in related contexts. Original proofs of the Mismatch Theorem and of the Matching Moment Property are given by using the properties of formal orthogonal polynomials and the Gauss quadrature for linear functionals.
View
Simplified anti-Gauss quadrature rules with applications in linear algebra
Article
Full-text available
  • Feb 2018
  • NUMER ALGORITHMS
The need to compute inexpensive estimates of upper and lower bounds for matrix functions of the form wTf(A)v with A∈Rn×nA\in {\mathbb {R}}^{n\times n} a large matrix, f a function, and v,w∈Rnv,w\in {\mathbb {R}}^{n} arises in many applications such as network analysis and the solution of ill-posed problems. When A is symmetric, u = v, and derivatives of f do not change sign in the convex hull of the spectrum of A, a technique described by Golub and Meurant allows the computation of fairly inexpensive upper and lower bounds. This technique is based on approximating vTf(A)v by a pair of Gauss and Gauss-Radau quadrature rules. However, this approach is not guaranteed to provide upper and lower bounds when derivatives of the integrand f change sign, when the matrix A is nonsymmetric, or when the vectors v and w are replaced by “block vectors” with several columns. In the latter situations, estimates of upper and lower bounds can be computed quite inexpensively by evaluating pairs of Gauss and anti-Gauss quadrature rules. When the matrix A is large, the dominating computational effort for evaluating these estimates is the evaluation of matrix-vector products with A and possibly also with AT. The calculation of anti-Gauss rules requires one more matrix-vector product evaluation with A and maybe also with AT than the computation of the corresponding Gauss rule. The present paper describes a simplification of anti-Gauss quadrature rules that requires the evaluation of the same number of matrix-vector products as the corresponding Gauss rule. This simplification makes the computational effort for evaluating the simplified anti-Gauss rule negligible when the corresponding Gauss rule already has been computed.
View
A Gauss/anti-Gauss quadrature method of moments applied to population balance equations with turbulence-induced nonlinear phase-space diffusion
Article
  • Jun 2022
  • J COMPUT PHYS
Many particulate systems occurring in nature and technology are adequately described by a number density function (NDF). The numerical solution of the corresponding population balance equation (PBE) is typically accompanied by high computational costs. Quadrature-based moment methods are an approach to reduce the computational complexity by solving only for a set of moments associated with the NDF employing Gaussian quadrature rules to close the moment equations derived from the PBE. The evolution of a population of inertial particles dispersed in a turbulent fluid is governed by a PBE with a phase-space diffusion term as a result of random microscale fluctuations. Considerations on the microscopic behavior concerning the momentum exchange between fluid and dispersed particles suggest that this diffusion term is nonlinear and nonsmooth. The resulting integral terms in the derived moment equations entail large approximation errors when using Gaussian quadrature rules for closure. In this work, we propose a modification of the quadrature method of moments (QMOM), namely the Gauss/anti-Gauss-QMOM (GaG-QMOM), making use of anti-Gaussian quadrature formulae to reduce the large errors due to this particular form of diffusion. This new method is investigated in a series of simple one-dimensional test cases with analytical reference solutions. Moreover, we propose a realizability-preserving variation of the strong-stability preserving Runge-Kutta (RKSSP) schemes that is suited to problems involving phase-space diffusion. Besides the observation that the modified second-order RKSSP scheme can serve as a realizability-preserving alternative to the standard RKSSP scheme of the same order, the numerical results reveal that, compared to the standard QMOM, the GaG-QMOM can reduce the large errors by one to two orders of magnitude.
View
Averaged Gauss quadrature formulas: Properties and applications
Article
  • Mar 2022
  • J COMPUT APPL MATH
The estimation of the quadrature error of a Gauss quadrature rule when applied to the approximation of an integral determined by a real-valued integrand and a real-valued nonnegative measure with support on the real axis is an important problem in scientific computing. Laurie developed anti-Gauss quadrature rules as an aid to estimate this error. Under suitable conditions the Gauss and associated anti-Gauss rules give upper and lower bounds for the value of the desired integral. It is then natural to use the average of Gauss and anti-Gauss rules as an improved approximation of the integral. Laurie also introduced these averaged rules. More recently, Spalević derived new averaged Gauss quadrature rules that have higher degree of exactness for the same number of nodes as the averaged rules proposed by Laurie. Numerical experiments reported in this paper show both kinds of averaged rules to often give much higher accuracy than can be expected from their degrees of exactness. This is important when estimating the error in a Gauss rule by an associated averaged rule. We use techniques similar to those employed by Trefethen in his investigation of Clenshaw–Curtis rules to shed light on the performance of the averaged rules. The averaged rules are not guaranteed to be internal, i.e., they may have nodes outside the convex hull of the support of the measure. This paper discusses three approaches to modify averaged rules to make them internal.
View
Orthogonal Polynomials: Computation and ApproximationComputation and Approximation
Book
  • Apr 2004
This is the first book on constructive methods for, and applications of orthogonal polynomials, and the first available collection of relevant Matlab codes. The book begins with a concise introduction to the theory of polynomials orthogonal on the real line (or a portion thereof), relative to a positive measure of integration. Topics which are particularly relevant to computation are emphasized. The second chapter develops computational methods for generating the coefficients in the basic three-term recurrence relation. The methods are of two kinds: moment-based methods and discretization methods. The former are provided with a detailed sensitivity analysis. Other topics addressed concern Cauchy integrals of orthogonal polynomials and their computation, a new discussion of modification algorithms, and the generation of Sobolev orthogonal polynomials. The final chapter deals with selected applications: the numerical evaluation of integrals, especially by Gauss-type quadrature methods, polynomial least squares approximation, moment-preserving spline approximation, and the summation of slowly convergent series. Detailed historic and bibliographic notes are appended to each chapter. The book will be of interest not only to mathematicians and numerical analysts, but also to a wide clientele of scientists and engineers who perceive a need for applying orthogonal polynomials.
View
Gauss quadrature for quasi-definite linear functionals
Article
  • Jul 2016
Gauss quadrature can be formulated as a method for approximating positive-definite linear functionals. Its mathematical context is extremely rich, with orthogonal polynomials, continued fractions and Padé approximation on one (functional analytic or approximation theory) side, and the method of moments,(real) Jacobi matrices, spectral decompositions and the Lanczos method on the other (algebraic) side. The quadrature concept can therefore be developed in many different ways. After a brief review of the mathematical interconnections in the positive-definite case, this paper will investigate the question of a meaningful generalization of Gauss quadrature for approximation of linear functionals that are not positive definite. For that purpose we use the algebraic approach, and, in order to build up the main ideas, recall the existing results presented in literature. Along the way we refer to the associated results expressed through the language of rational approximations. As the main result, we present the form of generalized Gauss quadrature and prove that the quasi-definiteness of the underlying linear functional represents a necessary and sufficient condition for its existence.
View
Calculation of Gaussian quadrature rules
Article
  • Jan 1969
Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus ∫abg(t)dt=∫abω(t)f(t)dt≈∑i=1Nwif(ti)\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i). Hopefully, the quadrature rule {wj,tj}j=1N{\{w_j, t_j\}}_{j=1}^{N} corresponding to the weight function ω\omega(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by ω\omega(t), and b) the moments of the weight function are known or can be calculated.
View