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Matrix Pencil Optimal Iterative Algorithms and Restarted Versions for Linear Matrix Equation and Pseudoinverse

Author

Listed:
  • Chein-Shan Liu

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Chung-Lun Kuo

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Chih-Wen Chang

    (Department of Mechanical Engineering, National United University, Miaoli 36063, Taiwan)

Abstract

We derive a double-optimal iterative algorithm (DOIA) in an m -degree matrix pencil Krylov subspace to solve a rectangular linear matrix equation. Expressing the iterative solution in a matrix pencil and using two optimization techniques, we determine the expansion coefficients explicitly, by inverting an m × m positive definite matrix. The DOIA is a fast, convergent, iterative algorithm. Some properties and the estimation of residual error of the DOIA are given to prove the absolute convergence. Numerical tests demonstrate the usefulness of the double-optimal solution (DOS) and DOIA in solving square or nonsquare linear matrix equations and in inverting nonsingular square matrices. To speed up the convergence, a restarted technique with frequency m is proposed, namely, DOIA(m); it outperforms the DOIA. The pseudoinverse of a rectangular matrix can be sought using the DOIA and DOIA(m). The Moore–Penrose iterative algorithm (MPIA) and MPIA(m) based on the polynomial-type matrix pencil and the optimized hyperpower iterative algorithm OHPIA(m) are developed. They are efficient and accurate iterative methods for finding the pseudoinverse, especially the MPIA(m) and OHPIA(m).

Suggested Citation

  • Chein-Shan Liu & Chung-Lun Kuo & Chih-Wen Chang, 2024. "Matrix Pencil Optimal Iterative Algorithms and Restarted Versions for Linear Matrix Equation and Pseudoinverse," Mathematics, MDPI, vol. 12(11), pages 1-31, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:11:p:1761-:d:1409467
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    References listed on IDEAS

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    1. Pan, V.Y. & Soleymani, F. & Zhao, L., 2018. "An efficient computation of generalized inverse of a matrix," Applied Mathematics and Computation, Elsevier, vol. 316(C), pages 89-101.
    2. Cordero, Alicia & Soto-Quiros, Pablo & Torregrosa, Juan R., 2021. "A general class of arbitrary order iterative methods for computing generalized inverses," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    3. R. H. AL-Obaidi & M. T. Darvishi & Predrag S. Stanimirović, 2022. "A Comparative Study on Qualification Criteria of Nonlinear Solvers with Introducing Some New Ones," Journal of Mathematics, Hindawi, vol. 2022, pages 1-20, September.
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