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In this paper we extend the Residual Arnoldi method for calculating anextreme eigenvalue (e.g. largest real part, dominant,...) to the case where thematrices depend on parameters. The difference between this Arnoldi method andthe classical Arnoldi algorithm is that in the former the residual is added tothe subspace. We develop a Tensor-Krylov method that applies the ResidualArnoldi method (RA) for a grid of parameter points at the same time. Thesubspace contains an approximate Krylov space for all these points. Instead ofadding the residuals for all parameter values to the subspace we create alow-rank approximation of the matrix consisting of these residuals and add onlythe 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 thesubspace has reached the prescribed maximal dimension. The novelty of thisapproach is twofold. Firstly, we observed that a large error in the low-rankapproximations is allowed without slowing down the convergence, which impliesthat we can do more iterations before restarting. Secondly, we pay particularattention to the way the subspace is restarted, since classical restartingtechniques give a too large subspace in our case. We motivate why it is goodenough to just keep the approximation of the searched eigenvector. At the endof the paper we extend this algorithm to shift-and-invert Residual Arnoldimethod to calculate the eigenvalue close to a shift $\sigma$ for a specificparameter dependency. We provide theoretical results and report numericalexperiments. The Matlab code is publicly available.

Tensor-Krylov method for computing eigenvalues of parameter-dependent matrices