The QR Algorithm for Eigenvalue Estimation: Theory and Experiments

Abstract

In this thesis, we explore one of the most important numerical algorithms ever invented, the QR algorithm for computing matrix eigenvalues. First, we describe out notations and mathematical symbols used throughout the thesis in Chapter 1. Then we lay the ground work by stating and proving some basic lemmas in Chapter 2. Then in Chapter 3, we prove the convergence of the QR algorithm under the assumption of distinct magnitudes for all eigenvalues. This constraint is relaxed in Chapter 4, where we prove the convergence of the QR algorithm under the assumption of possibly equal magnitude eigenvalues. Finally, in Chapter 5, we present some numerical experiments to validate the conclusions drawn in this thesis.