Extrapolation Methods for Vector Sequences 论文

摘要

This is an expository paper that describes and compares five methods for extrapolating to the limit (or anti-limit) of a vector sequence without explicit knowledge of the sequence generator. The methods are the minimal polynomial extrapolation (MPE) studied by Cabay and Jackson, Mešina, and Skelboe; the reduced rank extrapolation (RRE) of Eddy (which we show to be equivalent to Mešina’s version of MPE); the vector and scalar versions of the epsilon algorithm (VEA, SEA) introduced by Wynn and extended by Brezinski and Gekeler; and the topological epsilon algorithm (TEA) of Brezinski. We cover the derivation and error analysis of iterated versions of the algorithms, as applied to both linear and nonlinear problems, and we show why these versions tend to converge quadratically. We also present samples from extensive numerical testing that has led us to the following conclusions: (a) TEA, in spite of its role as a theoretical link between the polynomial-type and the epsilon-type methods, has no practical application; (b) MPE is at least as good as RRE, and VEA at least as good as SEA, in almost all situations (c) there are circumstances in which either MPE or VEA is superior to the other.