Error Bounds for the Lanczos Methods for Approximating Matrix Exponentials

Abstract

In this paper, we present new error bounds for the Lanczos method and the shift-and-invert Lanczos method for computing $e^{-\tau A} v$ for a large sparse symmetric positive semidefinite matrix $A$. Compared with the existing error analysis for these methods, our bounds relate the convergence to the condition numbers of the matrix that generates the Krylov subspace. In particular, we show that the Lanczos method will converge rapidly if the matrix $A$ is well-conditioned, regardless of what the norm of $\tau A$ is. Numerical examples are given to demonstrate the theoretical bounds.