New quadrature formulas based on the zeros of the Chebyshev polynomials of the second kind
type: Item
instance of: scholarly article
title: New quadrature formulas based on the zeros of the Chebyshev polynomials of the second kind
zbMATH Open document ID: 1073.65021
DOI: 10.1016/S0898-1221(03)00140-8
author: Shijun Yang
author: Xinghua Wang
publication date: 2004-08-06T00:00:00Z
review text: Quadrature formulas are derived for integrals \[ \int^{+1}_{-1} f(x)w(x)\,dx,\quad w(x) = (1 - x^2)^{1/2},\text{ or }w(x) = (1 - x^2)^{-1/2} \] which are accurate for polynomials of respective degree \(2(s+1)n +2s -1\), \(2(s+1)n+ 2s+1\), \(s,n\in\mathbb{N}\). The formulas employ divided differences at the zeros of \((1-x^2)Q_n(x)\) where \(Q_n(x)\) is an \(n\)-th degree Chebyshev polynomial of the second kind. When \(s = 0\) and \(n\) is replaced by \(2n -1\) the second formula reduces to a formula given by \textit{A. K. Varma} and \textit{E. Landau} [ibid. 30, No. 3--6, 213--220 (1995; Zbl 0833.41027)] exact for polynomials of degree \(4n -1\). The formulas are related to the Gauss-Turán quadrature formula.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 2082283
zbMATH Keywords: Quadrature formulas
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: Divided differences
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
cites work: New quadrature formulas based on the zeros of Jacobi polynomials
cites work: On weight functions which admit explicit Gauss-Turán quadrature formulas
cites work: On a quadrature formula of Micchelli and Rivlin
cites work: On some problems of P. Turán concerning \(L_m\) extremal polynomials and quadrature formulas
cites work: An analogue of problem 26 of P. Turán
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 2082283