Bounds on the error of fejer and clenshaw-curtis type quadrature for analytic functions

✍ Scribed by Paola Favati; Grazia Lotti; Francesco Romani

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 385 KB
• Volume: 6
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/0893-9659(93)90068-x

✦ Synopsis

We consider the problem of integrating a function f : [-1, l] -+ R which has an analytic extension f to an open disk Dr of radius r and center the origin, such that If(z)] 5 1 for any z E d,.

The goal of this paper is to study the minimal error among all algorithms which evaluate the integrand at the seros of the n-degree Chebyshev polynomials of first or second kind (F'ejer type quadrature formulas) or at the zeros of (n -2)-degree Chebyshev polynomials jointed with the endpoints -1,l (Clenshaw-Curtis type quadrature formulas), and to compare this error to the minimal error among all algorithms which evaluate the integrands at n points. In the case r > 1, it is easy to prove that Pejer and Clenshaw-Curtis type quadrature are almost optimal. In the case T = 1, we show that Fejer type formulas are not optimal since the error of any algorithm of this type is at least about ne2. These results hold for both the worst-case and the asymptotic settings.