TITLE:
Construction of Three Quadrature Formulas of Eighth Order and Their Application for Approximating Series
AUTHORS:
Bogusław Bożek, Wiesław Solak, Zbigniew Szydełko
KEYWORDS:
Quadrature and Cubature Formulas, Numerical Integration
JOURNAL NAME:
Applied Mathematics,
Vol.6 No.6,
June
9,
2015
ABSTRACT: In this paper, three types of three-parameters families of quadrature formulas for the Riemann’s integral on an interval of the real line are carefully studied. This research is a continuation of the results in the [1]-[3]. All these quadrature formulas are not based on the integration of an interpolant as so as the Gregory rule, a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order (see [4]). In some natural restrictions on the parameters we construct the only one quadrature formula of the eight order which belongs to the first, second and third family. For functions whose 8th derivative is either always positive or always negative, we use these quadrature formulas to get good two-sided bound on . Additionally, we apply these quadratures to obtain the approximate sum of slowly convergent series , where .