Numerical comparisons of nonlinear convergence accelerators

Abstract

As part of a continuing program of numerical tests of convergence accelerators, we have compared the iterated Aitken’s Δ 2 {\Delta ^2} method, Wynn’s ε \varepsilon algorithm, Brezinski’s θ \theta algorithm, and Levin’s u transform on a broad range of test problems: linearly convergence alternating, monotone, and irregular-sign series, logarithmically convergent series, power method and Bernoulli method sequences, alternating and monotone asymptotic series, and some perturbation series arising in applications. In each category either the ε \varepsilon algorithm or the u transform gives the best results of the four methods tested. In some cases differences among methods are slight, and in others they are quite striking.