Comparison of Birkhoff Type Quadrature Formulae

Abstract

Abstract. The classical approach to the theory of quadrature formulae is based on the concept of algebraic degree of precision (ADP). A quadrature formula ß, is considered to be "better" than Q2 if ADP(g, ) > ADP(ß2). However, there are many quadratures that use the same number of evaluations of the integrand and have the same ADP. Then, how should one compare such formulae? We show in this paper that the error of the quadrature depends monotonically on the type of data used. Roughly speaking, the lower the order of the derivatives used, the smaller is the error.As a consequence of the main result we demonstrate the existence of BirkhofF quadrature formulae of double precision.

On the Birkhoff Quadrature Formulas Using Even and Odd Order of Derivatives

“…In another paper, he showed in [11] that the formulas introduced in [9] are unique. Bojanov and Nikolov [12] showed that the error of the quadrature formulas depends monotonically on the data. Wang and Guo [13] obtained the asymptotic estimate of nodes and weights of Gaussian-Lobatto-Legendre-Birkhoff quadrature formulas.…”