The Radau-Lanczos Method for Matrix Functions

Analysis and development of restarted Krylov subspace methods for computing f(A)b have proliferated in recent years. We present an acceleration technique for such methods when applied to Stieltjes functions f and Hermitian positive definite matrices A. This technique is based on a rank-one modification of the Lanczos matrix derived from a connection between the Lanczos process and Gauss–Radau quadrature. We henceforth refer to the technique paired with the standard Lanczos method for matrix functions as the Radau–Lanczos method for matrix functions. We develop properties of general rank-one updates, leading to a framework through which other such updates could be explored in the future. We also prove error bounds for the Radau–Lanczos method, which are used to prove the convergence of restarted versions. We further present a thorough investigation of the Radau–Lanczos method explaining why it routinely improves over the standard Lanczos method. This is confirmed by several numerical experiments and we conclude that, in practical situations, the Radau–Lanczos method is superior in terms of iteration counts and timings, when compared to the standard Lanczos method.