We introduce the definition of tensorized block rational Krylov subspaces and its relation with multivariate rational functions, extending the formulation of tensorized Krylov subspaces introduced in [Kressner D., Tobler C., Krylov subspace methods for linear systems with tensor product structure, SIMAX, 2010]. 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.

翻译：本文引入了块有理Krylov张量子空间的定义及其与多元有理函数的关系，扩展了文献[Kressner D., Tobler C., Krylov subspace methods for linear systems with tensor product structure, SIMAX, 2010]中提出的Krylov张量子空间表述。此外，我们发展了基于张量块有理Krylov子空间投影的方法，用于求解具有低多线性秩或张量链秩的张量Sylvester方程。文中提供了收敛性分析、极点选择策略以及高效计算残差的技术。