Further convergence and stability results for the generalized Richardson extrapolation process GREP(1) with an application to the D(1)-transformation for in nite integrals

Let a(t) ∼ A+ ’(t)∑∞i=0 it as t → 0+, where a(t) and ’(t) are known for 0¡t6c for some c¿ 0, but A and the i are not known. The generalized Richardson extrapolation process GREP is used in obtaining good approximations to A, the limit or antilimit of a(t) as t → 0+. The convergence and stability properties of GREP for the case in which ’(t) ∼ t as t → 0+; 6= 0;−1;−2; : : : ; have been studied to a large extent in a recent work by the author. In the present work, we continue this study for the case in which is complex when the set of extrapolation points is {ti=t0!; i=0; 1; : : :} with !∈ (0; 1). We give a complete convergence and stability analysis under very weak assumptions on ’(t). We show that this analysis applies to the Levin–Sidi D-transformation that is a GREP, as this transformation is used for computing both convergent and divergent in nite-range integrals of functions f(x) that essentially satisfy f(x) ∼ x− −1 as x → ∞, with as above. In case of divergence, we show that the D-transformation produces approximations to the associated Hadamard nite parts. We append numerical examples that demonstrate the theory. c © 1999 Elsevier Science B.V. All rights reserved.