  In recent years, a great deal of attention has been focused on numerically
solving exponential integrators. The important ingredient to the implementation
of exponential integrators is the efficient and accurate evaluation of the so
called $\varphi$-functions on a given vector. The Krylov subspace method is an
important technique for this problem. For this type of method, however,
restarts become essential for the sake of storage requirements or due to the
growing computational complexity of evaluating the matrix function on a
Hessenberg matrix of growing size. Another problem in computing
$\varphi$-functions is the lack of a clear residual notion. The contribution of
this work is threefold. First, we introduce a framework of the harmonic Arnoldi
method for $\varphi$-functions, which is based on the residual and the oblique
projection technique. Second, we establish the relationship between the
harmonic Arnoldi approximation and the classical Arnoldi approximation, and
compare the harmonic Arnoldi method with the Arnoldi method from a theoretical
point of view. Third, we apply the thick-restarting strategy to the harmonic
Arnoldi method, and propose a thick-restated harmonic Arnoldi algorithm for
evaluating $\varphi$-functions. An advantage of the new algorithm is that we
can compute several $\varphi$-functions simultaneously in the same search
subspace. We show the merit of augmenting approximate eigenvectors in the
search subspace, and give insight into the relationship between the error and
the residual of $\varphi$-functions. Numerical experiments show the superiority
of our new algorithm over many state-of-the-art algorithms for the computation
of $\varphi$-functions.

Numerical Analysis