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 that 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 the 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 tr(f(A)) for a symmetric positive-definite matrix A by using the Lanczos method and make two contributions: (a) an error estimate for the bilinear form associated with f(A) and (b) 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.
Bibliographical noteFunding Information:
We are thankful to the anonymous referees whose comments help improve the paper. In particular, one referee pointed out an alternative derivation of the algorithm for computing the incremental error, mentioned at the end of Section 3.3. Jie Chen was supported by the XDATA program of the Defense Advanced Research Projects Agency (DARPA), administered through Air Force Research Laboratory contract FA8750-12-C-0323. Yousef Saad was supported by NSF Grant CCF-1318597.
XDATA program of the Defense Advanced Research Projects Agency (DARPA); Air Force Research Laboratory contract, Grant/Award Number: FA8750-12-C-0323; NSF, Grant/Award Number: CCF-1318597
Copyright © 2018 John Wiley & Sons, Ltd.
- Lanczos method
- confidence interval
- error estimate
- matrix function
- matrix trace