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An efficient computation of generalized inverse of a matrix

Author

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  • Pan, V.Y.
  • Soleymani, F.
  • Zhao, L.

Abstract

We propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves 18th order of convergence by using only seven matrix multiplications per iteration loop. This yields a high efficiency index for that computational task. The algorithm has a relatively mild numerical instability, and we stabilize it at the price of adding two extra matrix multiplications per iteration loop. This implies an efficiency index that exceeds the known record for numerically stable iterations for this task, which means substantial acceleration of the long standing algorithms for an important problem of numerical linear algebra. Our numerical tests cover a variety of examples in the category of generalized inverses, such as Drazin case, rectangular case, and preconditioning of linear systems. The test results are in good accordance with our formal study.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:316:y:2018:i:c:p:89-101
    DOI: 10.1016/j.amc.2017.08.010
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    Citations

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    Cited by:

    1. Xiaoji Liu & Naping Cai, 2018. "High-Order Iterative Methods for the DMP Inverse," Journal of Mathematics, Hindawi, vol. 2018, pages 1-6, May.
    2. Khosro Sayevand & Ahmad Pourdarvish & José A. Tenreiro Machado & Raziye Erfanifar, 2021. "On the Calculation of the Moore–Penrose and Drazin Inverses: Application to Fractional Calculus," Mathematics, MDPI, vol. 9(19), pages 1-23, October.
    3. Ma, Haifeng & Gao, Xiaoshuang & Stanimirović, Predrag S., 2020. "Characterizations, iterative method, sign pattern and perturbation analysis for the DMP inverse with its applications," Applied Mathematics and Computation, Elsevier, vol. 378(C).
    4. Kansal, Munish & Kumar, Sanjeev & Kaur, Manpreet, 2022. "An efficient matrix iteration family for finding the generalized outer inverse," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    5. 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.

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