Developing iterative algorithms to solve Sylvester tensor equations

Abstract This paper is concerned with solving high order Sylvester tensor equation arising in control theory. We propose the tensor forms of the bi-conjugate gradient and bi-conjugate residual methods for solving the tensor equation. To improve their performance, two preconditioned iterative algorithms based on the nearest Kronecker product are developed for finding its solution. We also prove that the proposed algorithms are convergent to an exact solution within finite iteration steps for any initial tensor in the absence of round-off errors. At last, some numerical examples are provided to illustrate the feasibility and validity of the algorithms proposed.

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