Preconditioned TBiCOR and TCORS algorithms for solving the Sylvester tensor equation.

• In Section 5 Preconditioned BiCOR and TCORS Algorithms are added, i.e., "PBiCOR" and "PTCORS", which are two main methods obtained by introducing the preconditioner based on the nearest Kronecker product in Loan and Pitsianis [28] to accelerate the proposed TBiCOR and TCORS methods in the previous version. • In "Numerical Experiments" section, we added "PBiCOR" and "PTCORS" and compared with other methods. Numerical examples show that PTCORS converges fastest among all methods listed, and PBiCOR converges a bit slower than PTCORS does. • We have added an acknowledgements part. Other sections, such as the title/ introduction/abstract/Numerical experiments/Conclusions section, are all updated. • We have checked the English usage of the revised paper carefully. In this paper, the preconditioned TBiCOR and TCORS methods are presented for solving the Sylvester tensor equation. A tensor Lanczos L -Biorthogonalization algorithm (TLB) is derived for solving the Sylvester tensor equation. Two improved TLB methods are presented. One is the biconjugate L -orthogonal residual algorithm in tensor form (TBiCOR), which implements the L U decomposition for the triangular coefficient matrix derived by the TLB method. The other is the conjugate L -orthogonal residual squared algorithm in tensor form (TCORS), which introduces a square operator to the residual of the TBiCOR algorithm. A preconditioner based on the nearest Kronecker product is used to accelerate the TBiCOR and TCORS algorithms, and we obtain the preconditioned TBiCOR algorithm (PTBiCOR) and preconditioned TCORS algorithm (PTCORS). The proposed algorithms are proved to be convergent within finite steps of iteration without roundoff errors. Several examples illustrate that the preconditioned TBiCOR and TCORS algorithms present excellent convergence. [ABSTRACT FROM AUTHOR]