Carl Jagels

A fast minimal residual algorithm for shifted unitary matrices

A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = ρU + ζI, where ρ and ζ are constants, U is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4] on the existence of conjugate gradient algorithms with short recurrence relations, and related results by Joubert and Young [13], can be extended.

On the construction of Szego polynomials

Abstract The Chebyshev and Stieltjes procedures are algorithms for computing recursion coefficients for polynomials that are orthogonal with respect to an inner product defined on (part of) the real axis. The Chebyshev procedure is an implementation of a map from moments to recursion coefficients of orthogonal polynomials. The modified Chebyshev procedure is an implementation of a map from modified moments to recursion coefficients. The latter map is generally much better conditioned than the former one. The conditioning of these maps has been studied by Gautschi. This paper is concerned with analogues of the Chebyshev and Stieltjes procedures when the inner product is defined on the unit circle. Polynomials orthogonal with respect to such an inner product are known as Szegő polynomials, and the analogue of the Chebyshev procedure is known as Schurs algorithm. This algorithm implements a map from moments to recursion coefficients for Szegő polynomials. Our analysis shows that this map generally is much better conditioned than the map implemented by the Chebyshev procedure, and suggests that Schurs algorithm is as insensitive to errors in the data as the modified Chebyshev procedure. Thus, roughly, moments associated with inner products defined on the unit circle correspond to modified moments associated with inner products defined on the real axis.

The structure of matrices in rational Gauss quadrature

This paper is concerned with the approximation of matrix functionals defined by a large, sparse or structured, symmetric definite matrix. These functionals are Stieltjes integrals with a measure supported on a compact real interval. Rational Gauss quadrature rules that are designed to exactly integrate Laurent polynomials with a fixed pole in the vicinity of the support of the measure may yield better approximations of these functionals than standard Gauss quadrature rules with the same number of nodes. Therefore it can be attractive to approximate matrix functionals by these rational Gauss rules. We describe the structure of the matrices associated with these quadrature rules, derive remainder terms, and discuss computational aspects. Also discussed are rational Gauss-Radau rules and the applicability of pairs of rational Gauss and Gauss-Radau rules to computing lower and upper bounds for certain matrix functionals.

Generalized averaged Szegź quadrature rules

Szegź quadrature rules are commonly applied to integrate periodic functions on the unit circle in the complex plane. However, often it is difficult to determine the quadrature error. Recently, Spalevic introduced generalized averaged Gauss quadrature rules for estimating the quadrature error obtained when applying Gauss quadrature over an interval on the real axis. We describe analogous quadrature rules for the unit circle that often yield higher accuracy than Szegź rules using the same moment information and may be used to estimate the error in Szegź quadrature rules.

On the computation of Gauss quadrature rules for measures with a monomial denominator

Let d µ be a nonnegative measure with support on the real axis and let α ? R be outside the convex hull of the support. This paper describes a new approach to determining recursion coefficients for Gauss quadrature rules associated with measures of the form d µ ? ( x ) : = d µ ( x ) / ( x - α ) 2 ? . The proposed method is based on determining recursion coefficients for a suitable family of orthonormal Laurent polynomials. Numerical examples show this approach to yield higher accuracy than available methods.