Variations of Zhang's Lanczos-Type Product Method
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Date
2000-12
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Report
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yes
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Abstract
Among the Lanczos-type product methods, which are characterized by residual polynomials $p_n t_n$ that are the product of the Lanczos polynomial $p_n$ and another polynomial $t_n$ of exact degree $n$ with $t_n(0) = 1$, Zhang's algorithm \GPBICG\ has the feature that the polynomials $t_n$ are implicitly built up by a pair of coupled two-term recurrences whose coefficients are chosen so that the new residual is minimized in a 2-dimensional space. There are several ways to achieve this. We discuss here alternative algorithms that are mathematically equivalent (that is, produce in exact arithmetic the same results). The goal is to find one where the ultimate accuracy of the iterates $x_n$ is guaranteed to be high and the cost is at most slightly increased.
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published
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Volume
2000-17
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Publisher
Seminar for Applied Mathematics, ETH Zurich
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Subject
Krylov space method; biconjugate gradients; Lanczos-type product method; BiCGxMR2; GPBi-CG
Organisational unit
02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
Notes
Dedicated to the memory of Rüdiger Weiss.