Extensions of Gauss, block Gauss, and Szego quadrature rules, with applications

This dissertation describes several new quadrature rules for the approximation of integrals determined by measures with support on the real axis or in the complex plane. Standard n-point Gauss rules are associated with symmetric tridiagonal matrices of order n. Averaged Gauss quadrature rules are obtained by 'flipping'; these tridiagonal matrices to obtain a quadrature rule of about twice the size. These averaged rules have been proposed by Spalevic. Gauss-type quadrature rules also can be defined when the measure has its support in the complex plane. These rules are associated with nonsymmetric tridiagonal matrices. This dissertation presents averaged Gauss-type quadrature rules associated with these Gauss-type quadrature rules. Also block extensions are described. These correspond to matrix-valued measures. Finally, averaged Szego quadrature rules are described. They extend standard Szego quadrature rules for the integration of function on the unit circle in the complex plane.