Numerical analysis is of importance in an increasing number of areas, owing in part to the ever - expanding role of computation in the mathematical sciences. Many algorithms in numerical analysis use polynomials orthogonal on (a subset of) a real interval. These algorithms include schemes for polynomial least squares approximation, Gaussian quadrature, and the Lanczos and conjugate gradient methods. It is planned to develop analogous algorithms that use polynomials orthogonal on the unit circle, also known as Szego polynomials. Szego polynomials are related to approximation by trigonometric polynomials, similarly as polynomials orthogonal on an interval are related to approximation by (algebraic) polynomials. Szego polynomials satisfy a recurrence relation with few terms. This makes them attractive to use in computations. They have already been applied successfully in algorithms for computing eigenvalues of unitary matrices, a problem that arises in certain frequency estimation methods in signal processing.
