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.