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Block-row and block-column iterative algorithms for solving linear matrix equation

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Abstract

Using tools like the Kronecker product and the vector operator, various Sylvester matrix equations can always be converted into the form \(Ax=f\), motivating us to investigate the algorithm for solving the linear matrix equation \(Ax=f\). By dividing the coefficient matrix A into row blocks or column blocks, a block-row iterative (BRI) algorithm, a block-column iterative (BCI) algorithm and an accelerated block-column iterative (ABCI) algorithm are developed to solve \(Ax=f\). It is successfully proved that the numerical solution produced by the proposed algorithms can converge to the exact solution for any given initial vector under appropriate conditions. Numerical examples are provided to demonstrate the effectiveness and superiority of the proposed algorithms.

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Acknowledgements

The authors cordially thank the editor and the anonymous referees for their helpful comments and suggestions which helped to improve the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, 100044, China

    Wenli Wang, Gangrong Qu & Duo Liu

  2. School of Mathematical Sciences, University of Jinan, Jinan, 250022, China

    Caiqin Song

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  1. Wenli Wang
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  2. Gangrong Qu
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  3. Caiqin Song
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  4. Duo Liu
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Correspondence to Gangrong Qu or Caiqin Song.

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The work is supported by National Natural Science Foundation of China (No. 61931003), Shandong Natural Science Foundation (Nos. ZR2020MA052, ZR2020MA055 and ZR2017BA025) and the Science and Technology Project of University of Jinan (No. XBS2008).

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Wang, W., Qu, G., Song, C. et al. Block-row and block-column iterative algorithms for solving linear matrix equation. Comp. Appl. Math. 42, 174 (2023). https://doi.org/10.1007/s40314-023-02312-y

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  • DOI: https://doi.org/10.1007/s40314-023-02312-y

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