Approximation of the linear combination of $\varphi$-functions using the block shift-and-invert Krylov subspace method

In this paper, we develop an algorithm in which the block shift-and-invert Krylov subspace method can be employed for approximating the linear combination of the matrix exponential and related exponential-type functions. Such evaluation plays a major role in a class of numerical methods known as exponential integrators. We derive a low-dimensional matrix exponential to approximate the objective function based on the block shift-and-invert Krylov subspace methods. We obtain the error expansion of the approximation, and show that the variants of its first term can be used as reliable a posteriori error estimates and correctors. Numerical experiments illustrate that the error estimates are efficient and the proposed algorithm is worthy of further study.