Abstract
This paper is concerned with computing approximations of matrix functionals of the form
F
(
A
)
:
=
v
T
f
(
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
.