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Krylov subspace methods for approximating a matrix function $f(A)$ times avector $v$ are analyzed in this paper. For the Arnoldi approximation to$e^{-\tau A}v$, two reliable a posteriori error estimates are derived from thenew bounds and generalized error expansion we establish. One of them is similarto the residual norm of an approximate solution of the linear system, and theother one is determined critically by the first term of the error expansion ofthe Arnoldi approximation to $e^{-\tau A}v$ due to Saad. We prove that each ofthe two estimates is reliable to measure the true error norm, and the secondone theoretically justifies an empirical claim by Saad. In the paper, byintroducing certain functions $\phi_k(z)$ defined recursively by the givenfunction $f(z)$ for certain nodes, we obtain the error expansion of theKrylov-like approximation for $f(z)$ sufficiently smooth, which generalizesSaad's result on the Arnoldi approximation to $e^{-\tau A}v$. Similarly, it isshown that the first term of the generalized error expansion can be used as areliable a posteriori estimate for the Krylov-like approximation to some othermatrix functions times $v$. Numerical examples are reported to demonstrate theeffectiveness of the a posteriori error estimates for the Krylov-likeapproximations to $e^{-\tau A}v$, $\cos(A)v$ and $\sin(A)v$.

A Posteriori Error Estimates of Krylov Subspace Approximations to Matrix Functions