Preconditioned TBiCOR and TCORS Algorithms for Solving the Sylvester Tensor Equation

TL;DR

This paper introduces preconditioned TBiCOR and TCORS algorithms for efficiently solving Sylvester tensor equations, utilizing tensor Lanczos biorthogonalization and preconditioning to enhance convergence and computational performance.

Contribution

The paper develops novel preconditioned TBiCOR and TCORS algorithms with proven finite-step convergence for Sylvester tensor equations, incorporating tensor Lanczos biorthogonalization and Kronecker product-based preconditioning.

Findings

Algorithms converge within finite steps without roundoff errors.

Preconditioning accelerates convergence significantly.

Numerical examples demonstrate superior performance and convergence.

Abstract

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.