Implicitly restarted Arnoldi with purification for the shift-invert transformation

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.