Szegő 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, Spalević 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 quadrat...

Approximations of matrix-valued functions of the form WT f(A)W , where A ∈ Rm×m is symmetric, W ∈ Rm×k , with m large and k m, has orthonormal columns, and f is a function, can be computed by applying a few steps of the symmetric block Lanczos method to A with initial block-vector W ∈ Rm×k . Golub and Meurant have shown that the approximants obtained in this manner may be considered block Gauss...

Abstract. Many problems in science and engineering require the evaluation of functionals of the form Fu(A) = uT f(A)u, where A is a large symmetric matrix, u a vector, and f a nonlinear function. A popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the Lanczos procedure applied to A with initial vector...

Orthogonal zero interpolants (OZI) are polynomials which interpolate the “zero-function” at a finite number of pre-assigned nodes and satisfy orthogonality condition. OZI’s can be constructed by the 3-term recurrence relation. These interpolants are found useful in the solution of constrained approximation problems and in the structure of Gauss-type quadrature rules. We present some theoretical...

In this paper, the evaluation of I = ∫ 1 −1 f(x) √ 1−x2 dx is proposed by using the opened and closed Gauss Chebyshev integration rules in the stochastic arithmetic. For this purpose, a theorem is proved to show the accuracy of the Gauss-Chebyshev rules. Then, the CESTAC 1 method and the stochastic arithmetic are used to validate the results and implement the numerical example.

Various concepts of orthogonality on the real line are reviewed that arise in connection with quadrature rules. Orthogonality relative to a positive measure and Gauss-type quadrature rules are classical. More recent types of orthogonality include orthogonality relative to a sign-variable measure, which arises in connection with Gauss-Kronrod quadrature, and power (or implicit) orthogonality enc...