Error Bounds for the Krylov Subspace Methods for Computations of Matrix Exponentials

In this paper, we present new a posteriori and a priori error bounds for the Krylov subspace methods for computing $e^{-\tau A}v$ for a given $\tau>0$ and $v \in C^n$, where $A$ is a large sparse non-Hermitian matrix. The {\em a priori} error bounds relate the convergence to $\lambda_{\min}\left(\frac{A+A^*}{2}\right)$, $\lambda_{\max}\left(\frac{A+A^*}{2}\right)$ (the smallest and the largest eigenvalue of the Hermitian part of $A$) and $|\lambda_{\max}\left(\frac{A-A^*}{2}\right)|$ (the largest eigenvalue in absolute value of the skew-Hermitian part of $A$), which define a rectangular region enclosing the field of values of $A$. In particular, our bounds explain an observed superlinear convergence behavior where the error may first stagnate for certain iterations before it starts to converge. The special case that $A$ is skew-Hermitian is also considered. Numerical examples are given to demonstrate the theoretical bounds.