  In this paper we extend the Residual Arnoldi method for calculating an
extreme eigenvalue (e.g. largest real part, dominant,...) to the case where the
matrices depend on parameters. The difference between this Arnoldi method and
the classical Arnoldi algorithm is that in the former the residual is added to
the subspace. We develop a Tensor-Krylov method that applies the Residual
Arnoldi method (RA) for a grid of parameter points at the same time. The
subspace contains an approximate Krylov space for all these points. Instead of
adding the residuals for all parameter values to the subspace we create a
low-rank approximation of the matrix consisting of these residuals and add only
the column space to the subspace. In order to keep the computations efficient,
it is needed to limit the dimension of the subspace and to restart once the
subspace has reached the prescribed maximal dimension. The novelty of this
approach is twofold. Firstly, we observed that a large error in the low-rank
approximations is allowed without slowing down the convergence, which implies
that we can do more iterations before restarting. Secondly, we pay particular
attention to the way the subspace is restarted, since classical restarting
techniques give a too large subspace in our case. We motivate why it is good
enough to just keep the approximation of the searched eigenvector. At the end
of the paper we extend this algorithm to shift-and-invert Residual Arnoldi
method to calculate the eigenvalue close to a shift $\sigma$ for a specific
parameter dependency. We provide theoretical results and report numerical
experiments. The Matlab code is publicly available.
