MATHEMATICS OF COMPUTATION Volume 66, Number 218, April 1997, Pages 667–689 S 0025-5718(97)00844-2 IMPLICITLY RESTARTED ARNOLDI WITH PURIFICATION FOR THE SHIFT-INVERT TRANSFORMATION KARL MEERBERGEN AND ALASTAIR SPENCE Abstract. The need to determine a few eigenvalues of a large sparse gen- eralised eigenvalue problem Ax = λBx 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 instabil- ities 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 com- ments on some extensions. 1. Introduction The problem of finding a few eigenvalues of large sparse N × N generalised eigenvalue problems of the form Ax = λBx , (1) with A nonsymmetric and B symmetric positive semidefinite, arises in many appli- cations. For example, the block structured eigenvalue problem  KC C T 0  u p  = λ  M 0 0 0  u p  , (2) with N = n + m, C ∈ R n×m of full rank, M ∈ R n×n positive definite appears in the stability analysis of steady state solutions of Stokes (K symmetric) and Navier- Stokes (K nonsymmetric) equations for incompressible flow, where u ∈ C n denotes the velocity component and p ∈ C m the pressure, see for example Cliffe, Garratt, and Spence [2]. Here M is the mass matrix of the velocity elements and K is nonsymmetric because of the linearisation of the convection term in the Navier- Stokes equations. As is well known, see Malkus [9], Ericsson [5] and Cliffe, Garratt, and Spence [3], (2) can have infinite eigenvalues, corresponding to eigenvectors of the form (0 T ,p T ) T . These have no physical relevance and in applications one would only be concerned about the calculation of a small number of ‘stability determining’ finite eigenvalues. A common approach for finding a few eigenvalues of (1) close to a given α ∈ C is Arnoldi’s method applied to the shift-invert transformation S =(A − αB) −1 B. Received by the editor May 9, 1995 and, in revised form, November 5, 1995. 1991 Mathematics Subject Classification. Primary 65F15, 65F50. Key words and phrases. Sparse generalised eigenvalue problems, shift-invert, semi-inner prod- uct, implicitly restarted Arnoldi. c 1997 American Mathematical Society 667