GAUSSIAN QUADRATURE FOR MATRIX VALUED FUNCTIONS ON THE UNIT CIRCLE

Abstract

. The Gaussian quadrature formulas for matrix valued functions on the unit circle are described. It is shown how the eigenvalues and eigenvectors of a unitary lower block Hessenberg matrix can be used to compute an approximation of a given matrix integral on the unit circle. A parallel algorithm for this purpose has been implemented on a IBM SP1 and some examples are worked out. Key words. orthogonal matrix polynomials, block Hessenberg matrices, quadrature, parallel algorithm. AMS subject classifications. 42C05, 41A55, 47A56, 65D32, 65Y05. 1. Introduction. The aim of this paper is to discuss the computation of Gaussian quadrature rules for matrix-valued functions on the unit circle on a high performance computer. These quadrature formulas generalize the scalar formulas considered in [12]. In [1], [10] and [11] the authors describe a divide and conquer method for the solution of the unitary Hessenberg eigenproblem, which is essentially the problem to be solved here. We will describe a ...