Simplified anti-Gauss quadrature rules with applications in linear algebra

The need to compute inexpensive estimates of upper and lower bounds for matrix functions of the form w T f ( A ) v with A∈ℝ^n× n a large matrix, f a function, and v,w∈ℝ^n arises in many applications such as network analysis and the solution of ill-posed problems. When A is symmetric, u = v , and derivatives of f do not change sign in the convex hull of the spectrum of A , a technique described by Golub and Meurant allows the computation of fairly inexpensive upper and lower bounds. This technique is based on approximating v T f ( A ) v by a pair of Gauss and Gauss-Radau quadrature rules. However, this approach is not guaranteed to provide upper and lower bounds when derivatives of the integrand f change sign, when the matrix A is nonsymmetric, or when the vectors v and w are replaced by “block vectors” with several columns. In the latter situations, estimates of upper and lower bounds can be computed quite inexpensively by evaluating pairs of Gauss and anti-Gauss quadrature rules. When the matrix A is large, the dominating computational effort for evaluating these estimates is the evaluation of matrix-vector products with A and possibly also with A T . The calculation of anti-Gauss rules requires one more matrix-vector product evaluation with A and maybe also with A T than the computation of the corresponding Gauss rule. The present paper describes a simplification of anti-Gauss quadrature rules that requires the evaluation of the same number of matrix-vector products as the corresponding Gauss rule. This simplification makes the computational effort for evaluating the simplified anti-Gauss rule negligible when the corresponding Gauss rule already has been computed.