| In this dissertation, we develop numerical methods and theories based on the relationship between orthogonal polynomials, quadrature rules, and linear algebra.;First, we analyze the sensitivity of Szego&huml; polynomials to the given moments. Also, we construct new quadrature rules related to Szego&huml; quadrature rules. The new quadrature rules are used to estimate the error in the Szego&huml; quadrature rules.;Next, we discuss the application of periodically restarted QR-algorithms for the computation of all eigenvalues of some structured matrices. This algorithm takes advantage of the structure of the matrices, while the QR-algorithm without restarts does not. As a result, the QR-algorithm with restarts becomes more efficient than the QR-algorithm without restarts for these matrices. We use the method for two types of structured matrices: rank-one modifications of Hermitian tridiagonal matrices and rank-one modifications of unitary upper Hessenberg matrices. The eigenvalues of the former are zeros of a linear combination of polynomials that satisfy a three-term recurrence relation and the eigenvalues of the latter matrices are zeros of Szego&huml; polynomials or a linear combination of Szego&huml; polynomials.;Finally, we introduce new quadrature rules based on the Arnoldi process. While the connection between the Lanczos process and Gaussian quadrature rules is well known, the connection between the Arnoldi process and quadrature has so far not received much attention. We construct Arnoldi quadrature rules and anti-Arnoldi quadrature rules using orthogonal polynomials with respect to bilinear forms defined by the Arnoldi process. |
