A Study of the Hr and Extended Hr Methods for the Standard Eigenvalue Problem

The QR method is a very eecient method for computing the spectrum of Hermitian tridiagonal matrices since the tridiagonal form is preserved over the iterations. For non-Hermitian tridiagonal matrices the QR method destroys the tridiagonal form. In this report we study two methods, the HR and the XHR methods, that preserve tridiagonal form for pseudo-Hermitian matrices. We also report results from numerical experiments using the algorithms and try to point out short comings and possible strategies for future improvement. Contents 1. Introduction 1 2. Orthogonality and rotational transformations 2 2.1. Unitary and Hermitian matrices and their real counterparts 2 2.2. Complex rotations 4 3. Decomposition type methods 9 3.1. Linear subspaces as a metric space 9