Matrix methods for quadrature formulas on the unit circle. A survey

In this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle. Like nodes and weights of Gauss quadrature formulas (for the estimation of integrals with respect to measures on the real line) can be computed from the eigenvalue decomposition of the Jacobi matrix, Szegő quadrature formulas (for the approximation of integrals with respect to measures on the unit circle) can be obtained from certain unitary five-diagonal or unitary Hessenberg matrices that characterize the recurrence for an orthogonal (Laurent) polynomial basis. These quadratures are exact in a maximal space of Laurent polynomials. Orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles. More general Szegő quadrature formulas can be obtained that are exact in certain spaces of rational functions. In this context, the nodes and the weights of these rules are computed from the eigenvalue decomposition of an operator Möbius transform of the same five-diagonal or Hessenberg matrices.