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T. Jawecki, W. Auzinger, O. Koch:
"Computable upper error bounds for Krylov approximations to matrix exponentials and associated phi-functions";
BIT Numerical Mathematics, 60 (2020), ISBN: 978-3-902627-11-7; 157 - 197.

An a posteriori estimate for the error of a standard Krylov approximation
to the matrix exponential is derived.
The estimate is based on the defect (residual) of the Krylov approximation and
is proven to constitute a rigorous upper bound on the error,
in contrast to existing asymptotical approximations.
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.
This means that the deviation of the error estimate from the true error
tends to zero faster than the error itself.
Furthermore, this result is extended to
Krylov approximations of phi-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.
Alternative 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 selectio

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

http://dx.doi.org/10.1007/s10543-019-00771-6

https://publik.tuwien.ac.at/files/publik_280454.pdf