  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.
