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T. Jawecki, W. Auzinger, O. Koch:
"Computable strict upper bounds for Krylov approximations to a class of matrix exponentials and φ-functions";
in: "ASC Report 23/2018", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2018, ISBN: 978-3-902627-11-7, 1 - 25.

An a posteriori bound for the error of a standard Krylov approximation to the exponential of nonexpansive matrices is derived. This applies for instance to skew-Hermitian matrices which appear in the numerical treatment of Schr¨odinger equations. It is proven that this defect-based bound is strict in contrast to existing approximations of the error, and it can be computed economically in the underlying Krylov space. In view of time-stepping applications, assuming that the given matrix is scaled by a time step, it is shown that the bound is asymptotically correct (with an order related to the dimension of the Krylov space) for the time step tending to zero. Furthermore, this result is extended to Krylov approximations of φ-functions and to improved versions of such approximations. The accuracy of the derived bounds is demonstrated by examples and compared with different variants known from the literature, which are also investigated more closely. Also other error bounds are tested on examples, in particular a version based on the concept of effective order. For the case where the matrix exponential is used in time integration algorithms, a step size selection strategy is proposed and illustrated by experiments.

matrix exponential · Krylov approximation · a posteriori error estimation · strict upper bound · Schr¨odinger equation

http://www.asc.tuwien.ac.at/preprint/2018/asc23x2018.pdf