Meerbergen, Karl; Spence, Alastair Implicitly restarted Arnoldi with purification for the shift-invert transformation. (English) Zbl 0864.65020 Math. Comput. 66, No. 218, 667-689 (1997). Summary: 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. Cited in 20 Documents MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65F50 Computational methods for sparse matrices Keywords:sparse generalised eigenvalue problems; shift-invert; semi-inner product; implicitly restarted Arnoldi method; numerical example Software:eigs PDF BibTeX XML Cite \textit{K. Meerbergen} and \textit{A. Spence}, Math. Comput. 66, No. 218, 667--689 (1997; Zbl 0864.65020) Full Text: DOI OpenURL