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

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

eigs
Full Text: