A Posteriori Error Estimates of Krylov Subspace Approximations to Matrix Functions

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