Many problems in science and engineering require the evaluation of functionals of
the form $F(A) = u^T f(A)u$, 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 $u$, and then evaluating pairs of Gauss and Gauss-Radau quadrature rules associated
with the tridiagonal matrix determined by the Lanczos procedure. The present paper extends this
approach to allow the use of rational Gauss quadrature rules.