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 Volume 7, Number 4, 2017, Pages 1402-1416                                                                DOI：10.11948/2017085 Approximation of the linear combination of $\varphi$-functions using the block shift-and-invert Krylov subspace method Dongping Li,Yuhao Cong Keywords:Matrix exponential, exponential integrators, block shift-and-invert Krylov subspace, a posteriori error estimates. Abstract: 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. PDF      Download reader