Search

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.