# Inexact inverse iteration and tuned preconditioning

## Melina Freitag

We consider the computation of an eigenvalue and corresponding eigenvector of a large sparse Hermitian positive definite matrix using inexact inverse iteration.

The large sparse systems arising at each iteration are often solved iteratively by means of preconditioned MINRES. We consider preconditioners based on incomplete Cholesky factorisation and show how the number of inner iterations increases per outer iteration compared to the non-preconditioned case. A tuned preconditioner is derived which shows considerable improvement over the standard preconditioner. Furthermore we compare the spectral properties of the standard preconditioned matrix with those of the tuned preconditioned matrix.