In this paper, the preconditioned TBiCOR and TCORS methods are presented for solving the Sylvester tensor equation. A tensor Lanczos $\mathcal{L}$-Biorthogonalization algorithm (TLB) is derived for solving the Sylvester tensor equation. Two improved TLB methods are presented. One is the biconjugate $\mathcal{L}$-orthogonal residual algorithm in tensor form (TBiCOR), which implements the $LU$ decomposition for the triangular coefficient matrix derived by the TLB method. The other is the conjugate $\mathcal{L}$-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.

翻译：本文介绍了解决Sylvester Exmoncor 等式的先决条件 TBCOR 和 TCORS 的方法。 一种是用于解决 Sylvester Exmoncor 等式的“ 软体”, 一种是用于解决 Sylvester Exmocor 等式的“ 软体” 和 TCOR 等式的“ 软体” 和“ 软体” 的“ 软体” 方法。 另一种是用于解决 Sylvester Exmor 等式的“ 软体” 的“ 软体” 朗克” 和“ 软体” 的“ 软体” 方法。 另一种是用于解决Sylvester Exquald 等式的“ 软体” 的“ 软体” 软体“ 软体” 。 两种改进的TLBCOR 和“ 软体” 的“ 软体” 后项算法。, 我们获得了由TBCOR 方法衍生的“ 和“ 立体” 的“ 后项法” 的“ 和“ 后项法” 。 的“ 的“ 的“ 后项” 的“ 的“ 后项法” 的“ 。