Iterations of convergence accelerating nonlinear transforms
Abstract
The role of iteration, or the repeated application of an algorithm in accelerating the convergence of sequences, is investigated for certain wellknown nonlinear transforms (like the ∊, u, θ and ϱ). It is found that the iterations of the Levin u transform improve upon the stability of the direct transform and are successful over a wide class of alternating, monotone and divergent series. An estimate of the numerical error in using iterations of the u_{2} (or θ _{2}) transform is given. A comparison of the estimated error with the actual error shows that the iterations of the u_{2} transform coupled with this error estimate provides a uniformly successful and, for most purposes, practical method of evaluating the limit of a sequence. A comparison of this method with other direct methods and their iterations is given over a wide class of alternating, monotone and divergent series.
 Publication:

Computer Physics Communications
 Pub Date:
 April 1989
 DOI:
 10.1016/00104655(89)900301
 Bibcode:
 1989CoPhC..54...31B