Planar-CG methods and Matrix Tridiagonalization in Large scale Unconstrained Optimization

In this paper we aim at carrying out and describing some issues for real eigenvalue computation via iterative methods. More specifically we work out new techniques for iteratively developing specific tridiagonalizations of a {\em symmetric} and {\em indefinite} matrix $A \in \re^{n \times n}$, by means of suitable Krylov subspace algorithms defined in \cite{13}, \cite{21}. These schemes represent extensions of the well known Conjugate Gradient (CG) method to the indefinite case. We briefly recall these algorithms and we suggest a comparison with the method in \cite{18}, along with a discussion on the practical application of the proposed results for eigenvalue computation. Furthermore, we focus on motivating the fruitful use of these tridiagonalizations for ensuring the convergence to second order points, within an optimization framework.