Error estimates for Gaussian quadratures of analytic functions

• Published in: Journal of computational and applied mathematics Vol. 233; no. 3; pp. 802 - 807
• Main Authors: MILOVANOVIC, Gradimir V, SPALEVIC, Miodrag M, PRANIC, Miroslav S
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Kidlington Elsevier B.V 01.12.2009; Elsevier
• Subjects: Approximations and expansions; Chebyshev weight function; Contour integral representation; Error bound; Exact sciences and technology; Functions of a complex variable; Gaussian quadrature formula; Mathematical analysis; Mathematics; Measure and integration; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Remainder term for analytic functions; Sciences and techniques of general use; secondary; Remainder term for analytic functions; Chebyshev weight function; Gaussian quadrature formula; Error bound; Contour integral representation; primary; Numerical integration; Error estimation; Numerical approximation; primary 65D30; Numerical analysis; Cubature; Applied mathematics; secondary 41A55; Weight function; 65D32; Quadrature formula; Integral representation; Analytical function
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2009.02.048

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points ±1 and the sum of semi-axes ϱ > 1 for the Chebyshev weight functions of the first, second and third kind, and derive representation of their difference. Using this representation and following Kronrod’s method of obtaining a practical error estimate in numerical integration, we derive new error estimates for Gaussian quadratures.