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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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Keywords: Acceleration of convergence, iterated Aitken’s <!– MATH ${\Delta ^2}$ –> <IMG WIDTH="31" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${\Delta ^2}$">, <!– MATH $\varepsilon$ –> <IMG WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$\varepsilon$"> algorithm, <IMG WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img7.gif" ALT="$\theta$"> 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