Monotonicity in Romberg quadrature

Abstract

Monotonicity of one or more derivatives of the integrand is shown to imply a corresponding property of the approximating Romberg scheme. This is of importance in connection with error estimation by majorants [6]. The monotonicity properties are derived from an elementary study of the kernel functions involved. A possible explanation is given of the monotonicity which frequently occurs in applications where nothing is presupposed about the signs of derivatives of the integrand. Finally, a nonlinear addition to Havie’s scheme is suggested.