New convergence results on the generalized Richardson extrapolation process GREP (1) for logarithmic sequences

• Authors: Avram Sidi
• Venue: Mathematics of Computation
• Year: 2002

Let a(t) ∼ A + φ(t) Σi=0∞ βiti as t → 0+, where a(t) and φ(t) are known for 0 t ≤ c for some c 0, 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 approximations An(j) to A obtained via GREP(1) are defined by the linear systems a(tl) = An(j) + φ(tl) Σi=0n-1 β'itli, l=j,j+1,...,j + n, where {tl}l=0∞ is a decreasing positive sequence with limit zero. The study of GREP(1) for slowly varying functions a(t) was begun in two recent papers by the author. For such a(t) we have φ(t) ∼ αtδ as t → 0+ with δ possibly complex and δ ≠ 0, -1, -2,.... In the present work we continue to study the convergence and stability of GREP(1) as it is applied to such a(t) with different sets of collocation points tl that have been used in practical situations. In particular, we consider the cases in which (i) tl are arbitrary, (ii) liml→∞ tl+1/tl = 1, (iii) tl ∼ cl-q as l → ∞ for some c,q 0, (iv) tl+1/tl ≤ ω ∈ (0, 1) for all l, (v) liml → ∞ tl+1/tl = ω ∈ (0, 1), and (vi) tl+1/tl= ω ∈ (0, 1) for all l.