On a Generalization of the Richardson Extrapolation Process

A convergence result for a generalized Richardson extrapolation process is improved upon considerably and additional results of interest are proved. An application of practical importance is also given. Finally, some known results concerning the convergence of Levin's T-transformation are reconsidered in light of the results of the present work. Let A(y) be a scalar function of a discrete or continuous variable y, defined for 0 < y =< b < ~. Let there exist constants A and ~k, k = 1, 2 .... , and functions 4~k (Y), k = 1, 2 ..... which form an asymptotic sequence in the sense that (1.1) C~k+l(y)=o(d~k(y)) as y~O+, and assume that A (y) has the asymptotic expansion (1.2) A(y)..~A+ ~kC~k(y) as y~0+. k=l Here A(y) and ~k(Y), k=l, 2, ..., are assumed to be known for 0<y=<b, but A and ak, k = 1, 2 ..... are unknown. The problem is to approximate A, which, in many cases is lim A(y) when the latter exists. (When lim A(y) does not y-*O+ y~O+ exist, A is said to be the antilimit of A(y) as y ~ 0+ .)