Rational gauss quadrature rules for the approximation of matrix functionals involving stieltjes functions

This paper is concerned with computing approximations of matrix functionals of the form F(A) : = v^Tf(A) v, where A is a large symmetric positive definite matrix, v is a vector, and f is a Stieltjes function. We approximate F(A) with the aid of rational Gauss quadrature rules. Associated rational Gauss–Radau and rational anti-Gauss rules are developed. Pairs of rational Gauss and rational Gauss–Radau quadrature rules, or pairs of rational Gauss and rational anti-Gauss quadrature rules, can be used to determine upper and lower bounds, or approximate upper and lower bounds, for F(A). The application of rational Gauss rules, instead of standard Gauss rules, is beneficial when the function f has singularities close to the spectrum of A.