The Richardson Extrapolation Process with a Harmonic Sequence of Collocation Points

Let A(y) ∼ A+ ∑∞ k=1 αky σk as y → 0+, where y is a discrete or continuous variable. Assume that σk are known numbers that may be complex in general and that A(y) is known for y ∈ (0, b] for some b > 0. The aim is to find or approximate A, the limit or antilimit of A(y) as y → 0+. One very effective way of approximating A is by the Richardson extrapolation process that is defined via the linear systems A(yl) = A (j) n + ∑n k=1 αky σk l , j ≤ l ≤ j + n. Here A n are approximations to A and αk are additional unknowns. The yl are picked such that y0 > y1 > y2 > · · · > 0 and liml→∞ yl = 0. In this paper we give a detailed analysis of the convergence and stability of the column sequences {A n } ∞ j=0 with n fixed, when yl = c/(l + η) q for some positive c, η, and q. Specifically, we prove that convergence takes place as j → ∞ and give the precise rate at which it does. We also prove that the process is unstable and quantify its instability asymptotically. This instability may be dealt with numerically by using high-precision floating-point arithmetic.