Computation of error bounds via generalized Gauss–Radau and Gauss–Lobatto rules

Many functionals of a large symmetric matrix of interest in science and engineering can be expressed as a Stieltjes integral with a measure supported on the real axis. These functionals can be approximated by quadrature rules. Golub and Meurant proposed a technique for computing upper and lower error bounds for Stieltjes integrals with integrands whose derivatives do not change sign on the convex hull of the support of the measure. This technique is based on evaluating pairs of a Gauss quadrature rule and a suitably chosen Gauss–Radau or Gauss–Lobatto quadrature rule. However, when derivatives of the integrand change sign on the convex hull of the support of the measure, this technique is not guaranteed to give upper and lower error bounds for the functional. We describe an extension of the technique by Golub and Meurant that yields upper and lower error bounds for the functional in situations when only some derivatives of the integrand do not change sign on the convex hull of the support of the measure. This extension is based on the use of pairs of Gauss, and suitable generalized Gauss–Radau or Gauss–Lobatto rules. New methods to evaluate generalized Gauss–Radau and Gauss–Lobatto rules also are described.