Abstract
Starting from the explicit expression of the corresponding kernels, derived by Gautschi and Li (W. Gautschi, S. Li: The remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadrature rules with multiple end points, J. Comput. Appl. Math. 33 (1990) 315{329), we determine the exact dimensions of the minimal ellipses on which the modulus of the kernel starts to behave in the described way. The effective error bounds for Gauss- Radau and Gauss-Lobatto quadrature formulas with double end point(s) are derived. The comparisons are made with the actual errors.
Highlights
Gautschi and Lis conjectures on Gauss-Lobatto quadratures with Chebyshev weight functions of the third and the fourth kind were already proved in [6]
Those cases required a simpler analysis compared to the cases addressed here
[3] Gautschi and Li analyzed the maximum modulus of the kernel with KnR,r(z; ω3)
Summary
We analyze the remainder term of Gauss-Radau quadrature rule with the end point −1 of multiplicity r, r−1 n (1). If the integrand f is analytic function in a domain D containing [−1, 1], the remainder terms RnR,,rL(f ) admit the contour integral representation (3). In [3] Gautschi and Li considered Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points with respect to the four Chebyshev weight functions ω1(t). Gautschi and Lis conjectures on Gauss-Lobatto quadratures with Chebyshev weight functions of the third and the fourth kind were already proved in [6]. Those cases required a simpler analysis compared to the cases addressed here. For error bounds of quadrature rules for analytic functions see the recent survey paper by Notaris [8]
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