Tensorized block rational Krylov methods for tensor Sylvester equations

We introduce the definition of tensorized block rational Krylov subspace and its relation with multivariate rational functions, extending the formulation of tensorized Krylov subspaces introduced in Kressner, D. & Tobler, C. (2010) Krylov subspace methods for linear systems with tensor product structure. SIAM J. Matrix Anal. Appl., 31,$1688$–$1714$. Moreover, we develop methods for the solution of tensor Sylvester equations with low multilinear or tensor train rank, based on projection onto a tensor block rational Krylov subspace. We provide a convergence analysis, some strategies for pole selection and techniques to efficiently compute the residual.