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

摘要

Let a(t)∼A+ϕ(t)∑i=0∞βiti as t→0+, where a(t) and ϕ(t) are known for 00, but A and the βi are not known. The generalized Richardson extrapolation process GREP(1) 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(1) for the case in which ϕ(t)∼αtδ as t→0+,δ≠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 δ is complex when the set of extrapolation points is {ti=t0ωi,i=0,1,…} with ω∈(0,1). We give a complete convergence and stability analysis under very weak assumptions on ϕ(t). We show that this analysis applies to the Levin–Sidi D(1)-transformation that is a GREP(1), as this transformation is used for computing both convergent and divergent infinite-range integrals of functions f(x) that essentially satisfy f(x)∼νx−δ−1 as x→∞, with δ as above. In case of divergence, we show that the D(1)-transformation produces approximations to the associated Hadamard finite parts. We append numerical examples that demonstrate the theory.