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An outstanding problem when computing a function of a matrix, $f(A)$, byusing a Krylov method is to accurately estimate errors when convergence isslow. Apart from the case of the exponential function which has beenextensively studied in the past, there are no well-established solutions to theproblem. Often the quantity of interest in applications is not the matrix$f(A)$ itself, but rather, matrix-vector products or bilinear forms. When thecomputation 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 errorestimates is that the accuracy of the computed result is unknown. In thispaper, we consider the problem of computing $\mathrm{tr}(f(A))$ for a symmetricpositive-definite matrix $A$ by using the Lanczos method and make twocontributions: (i) we propose an error estimate for the bilinear formassociated with $f(A)$, and (ii) an error estimate for the trace of $f(A)$. Wedemonstrate the practical usefulness of these estimates for large matrices andin particular, show that the trace error estimate is indicative of the numberof accurate digits. As an application, we compute the log-determinant of acovariance matrix in Gaussian process analysis and underline the importance oferror tolerance as a stopping criterion, as a means of bounding the number ofLanczos steps to achieve a desired accuracy.

A Posteriori Error Estimate for Computing $\mathrm{tr}(f(A))$ by Using the Lanczos Method