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Preconditioned TBiCOR and TCORS algorithms for solving the Sylvester tensor equation

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  • Huang, Guang-Xin
  • Chen, Qi-Xing
  • Yin, Feng

Abstract

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 LU 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.

Suggested Citation

  • Huang, Guang-Xin & Chen, Qi-Xing & Yin, Feng, 2022. "Preconditioned TBiCOR and TCORS algorithms for solving the Sylvester tensor equation," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    • Handle: RePEc:eee:apmaco:v:422:y:2022:i:c:s0096300322000704
    DOI: 10.1016/j.amc.2022.126984
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    References listed on IDEAS

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    1. Zhang, Xin-Fang & Wang, Qing-Wen, 2021. "Developing iterative algorithms to solve Sylvester tensor equations," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    2. Zhen Chen & Linzhang Lu, 2013. "A Gradient Based Iterative Solutions for Sylvester Tensor Equations," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-7, March.
    3. Lv, Changqing & Ma, Changfeng, 2020. "A modified CG algorithm for solving generalized coupled Sylvester tensor equations," Applied Mathematics and Computation, Elsevier, vol. 365(C).
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    Cited by:

    1. Chen, Qi-Xing & Huang, Guang-Xin & Zhang, Ming-Yue, 2024. "Preconditioned BiCGSTAB and BiCRSTAB methods for solving the Sylvester tensor equation," Applied Mathematics and Computation, Elsevier, vol. 466(C).

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