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Implicitly restarted Arnoldi with purification for the shift-invert transformation

Authors: Karl Meerbergen and Alastair Spence
Journal: Math. Comp. 66 (1997), 667-689
MSC (1991): Primary 65F15, 65F50
MathSciNet review: 1408376
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Abstract: The need to determine a few eigenvalues of a large sparse generalised eigenvalue problem $Ax=\lambda Bx$ with positive semidefinite $B$ arises in many physical situations, for example, in a stability analysis of the discretised Navier-Stokes equation. A common technique is to apply Arnoldi's method to the shift-invert transformation, but this can suffer from numerical instabilities as is illustrated by a numerical example. In this paper, a new method that avoids instabilities is presented which is based on applying the implicitly restarted Arnoldi method with the $B$ semi-inner product and a purification step. The paper contains a rounding error analysis and ends with brief comments on some extensions.

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Additional Information

Karl Meerbergen
Affiliation: LMS Numerical Technologies, Interleuvenlaan 70, 3001 Heverlee, Belgium

Alastair Spence
Affiliation: School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

Keywords: Sparse generalised eigenvalue problems, shift-invert, semi-inner product, implicitly restarted Arnoldi.
Received by editor(s): May 9, 1995
Received by editor(s) in revised form: November 5, 1995
Article copyright: © Copyright 1997 American Mathematical Society