Monotonicity in Romberg Quadrature

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. Following Bauer, Rutishauser, and Stiefel [1] (B.R.S.) and Hävie [2] we denote by ro" and I/o*' the approximations to jl f(x) dx by the trapezoidal and the rectangular rules with step sizes 2~*, k = 0, 1, • ■ • . Repeated Richardson extrapolation leads to the ordinary and modified Romberg schemes and U1^ satisfying (1) Tlk> = r£LV' + (TLk:r Tl£Sh*m i). (2) Ul» = j£. We may form an aggregate table illustrated in Fig. 1.