adaptive simpler gmres based on tensor format for sylvester tensor equation

The problem of sylvester tensor equations is a crucial issue inseveral research applications. krylov subspace methods are very eectiveapproaches to solve this problems due to their merits in large and sparseproblems. we present an adaptive simpler gmres method for solving thesylvester tensor equation and then obtain an upper bound for condition num-ber of the basis matrix. eventually, a numerical example is conducted toillustrate the eectiveness of the method.