  An outstanding problem when computing a function of a matrix, $f(A)$, by
using a Krylov method is to accurately estimate errors when convergence is
slow. Apart from the case of the exponential function which has been
extensively studied in the past, there are no well-established solutions to the
problem. Often the quantity of interest in applications is not the matrix
$f(A)$ itself, but rather, matrix-vector products or bilinear forms. When the
computation related to $f(A)$ is a building block of a larger problem (e.g.,
approximately computing its trace), a consequence of the lack of reliable error
estimates is that the accuracy of the computed result is unknown. In this
paper, we consider the problem of computing $\mathrm{tr}(f(A))$ for a symmetric
positive-definite matrix $A$ by using the Lanczos method and make two
contributions: (i) we propose an error estimate for the bilinear form
associated with $f(A)$, and (ii) an error estimate for the trace of $f(A)$. We
demonstrate the practical usefulness of these estimates for large matrices and
in particular, show that the trace error estimate is indicative of the number
of accurate digits. As an application, we compute the log-determinant of a
covariance matrix in Gaussian process analysis and underline the importance of
error tolerance as a stopping criterion, as a means of bounding the number of
Lanczos steps to achieve a desired accuracy.
