Implicitly restarted Arnoldi with purification for the shift-invert transformation. (English) Zbl 0864.65020

Meerbergen, Karl; Spence, Alastair

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.

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