Estimating the condition number of f(A)b

Abstract

New algorithms are developed for estimating the condition number of f(A)b, where A is a matrix and b is a vector. The condition number estimation algorithms for f(A) already available in the literature require the explicit computation of matrix functions and their Fr´echet derivatives and are therefore unsuitable for the large, sparse A typically encountered in f(A)b problems. The algorithms we propose here use only matrix-vector multiplications. They are based on a modified version of the power iteration for estimating the norm of the Fr´echet derivative of a matrix function, and work in conjunction with any existing algorithm for computing f(A)b. The number of matrix-vector multiplications required to estimate the condition number is proportional to the square of the number of matrix-vector multiplications required by the underlying f(A)b algorithm. We develop a specific version of our algorithm for estimating the condition number of e A b, based on the algorithm of Al-Mohy and Higham (SIAM J. Matrix Anal. Appl. 30(4), 1639–1657, 2009). Numerical experiments demonstrate that our condition estimates are reliable and of reasonable cost.