Connections Between Lanczos Iteration and Orthogonal Polynomials

Abstract

In this thesis we examine the connections between orthogonal polynomials and the Lanczos algorithm for tridiagonalizing a Hermitian matrix. The Lanczos algorithm provides an easy way to calculate and to estimate the eigenvalues and eigenvectors of such a matrix. It also forms the basis of several popular iterative methods for solving linear systems of the form $Ax = b$, where $A$ is an $m \times m$ Hermitian matrix and $b$ is an $m \times 1$ column vector. Iterative methods often provide significant computational savings when solving such systems.

We demonstrate how the Lanczos algorithm gives rise to a three-term recurrence, from which a family of orthogonal polynomials may be derived. We explore two of the more important consequences of this line of thought: the behavior of the Lanczos iteration in the presence of finite-precision arithmetic, and the ability of the Lanczos iteration to compute zeros of orthogonal polynomials. A deep understanding of the former is crucial to actual software implementation of the algorithm, while knowledge of the latter provides an easy and efficient means of constructing quadrature rules for approximating integrals.