Two New Classes of Nonlinear Transformations for Accelerating the Convergence of Infinite Integrals and Series

Abstract

Two new classes of nonlinear transformations, the D-transformation to accelerate the convergence of infinite integrals and the d-transformation to accelerate the convergence of infinite series, are presented. In the course of the development of these transformations two interesting asymptotic expansions, one for infinite integrals and the other for infinite series, are derived. The transformations D and d can easily be applied to infinite integrals ∫ ∞ 0 f( t) dt whose integrands f( t) satisfy linear differential equations of the form f( t)=Σ m k=1 p k ( t) f k ( t) and to infinite series Σ ∞ r=1 f( r) w terms f( r) satisfy a linear difference equation of the form f( r)=Σ m k=1 p k ( r)Δ k f( r), such that in both cases the p k have asymptotic expansions in inverse powers of their arguments. In order to be able to apply these transformations successfully one need not know explicitly the differential equation that the integrand satisfies or the difference equation that the terms of the series satisfy; mere knowledge of the existence of such a differential or difference equation and its order m is enough. This broadens the areas to which these methods can be applied. The connection between the D- and d-transformations with some known transformations in shown. The use and the remarkable efficiency of the D- and d-transformations are demonstrated through several numerical examples. The computational aspects of these transformations are described in detail.