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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Numerical comparisons of nonlinear convergence accelerators


Authors: David A. Smith and William F. Ford
Journal: Math. Comp. 38 (1982), 481-499
MSC: Primary 65B10; Secondary 65-04
MathSciNet review: 645665
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Abstract: As part of a continuing program of numerical tests of convergence accelerators, we have compared the iterated Aitken's $ {\Delta ^2}$ method, Wynn's $ \varepsilon $ algorithm, Brezinski's $ \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 $ \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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1982-0645665-1
PII: S 0025-5718(1982)0645665-1
Keywords: Acceleration of convergence, iterated Aitken's $ {\Delta ^2}$, $ \varepsilon $ algorithm, $ \theta $ algorithm, Levin's transforms, linear convergence, logarithmic convergence, power series, Fourier series, power method, Bernoulli's method, asymptotic series, perturbation series, numerical tests
Article copyright: © Copyright 1982 American Mathematical Society