%AChen, Jie [IBM Thomas J. Watson Research Center Yorktown Heights NY USA]%AChen, Jie [IBM Thomas J. Watson Research Center; Yorktown Heights NY USA]%ASaad, Yousef [Department of Computer Science and Engineering University of Minnesota Twin Cities Minneapolis USA]%ASaad, Yousef [Department of Computer Science and Engineering; University of Minnesota Twin Cities; Minneapolis USA]%BJournal Name: Numerical Linear Algebra with Applications; Journal Volume: 25; Journal Issue: 5; Related Information: CHORUS Timestamp: 2024-07-01 21:21:15
%D2018%IWiley Blackwell (John Wiley & Sons)
%JJournal Name: Numerical Linear Algebra with Applications; Journal Volume: 25; Journal Issue: 5; Related Information: CHORUS Timestamp: 2024-07-01 21:21:15
%K
%MOSTI ID: 10056728
%PMedium: X
%TA posteriori error estimate for computing tr(
*f*
(
*A*
)) by using the Lanczos method
%XSummary
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 matrixf(A) itself but rather the matrix–vector products or bilinear forms. When the computation related tof(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 matrixAby using the Lanczos method and make two contributions: (a) an error estimate for the bilinear form associated withf(A) and (b) an error estimate for the trace off(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.

%0Journal Article