Search Advanced

Electronic Research Archive

2024, Volume 32, Issue 9: 5306-5320. doi: 10.3934/era.2024245
Previous Article Next Article
Research article

A structure-preserving doubling algorithm for the square root of regular M-matrix

  • 1.
    School of Mathematics, Taiyuan University of Technology, Jinzhong 030600, China
  • 2.
    School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong 030619, China
  • 3.
    Institute of Mathematics and Computer Science, University of Sindh, Jamshoro, Pakistan
  • Received: 10 May 2024 Revised: 27 August 2024 Accepted: 06 September 2024 Published: 18 September 2024

  • The matrix square root is widely encountered in many fields of mathematics. In this paper, based on the properties of M-matrix and quadratic matrix equations, we study the square root of M-matrix, and prove that for a regular M-matrix there always exists a regular M-matrix as its square root. In addition, a structure-preserving doubling algorithm is proposed to compute the square root. Theoretical analysis and numerical experiments are given to show that our method is feasible and is effective under certain conditions.

    Citation: Zehua Wang, Jinrui Guan, Ahmed Zubair. A structure-preserving doubling algorithm for the square root of regular M-matrix[J]. Electronic Research Archive, 2024, 32(9): 5306-5320. doi: 10.3934/era.2024245

    Related Papers:

  • The matrix square root is widely encountered in many fields of mathematics. In this paper, based on the properties of M-matrix and quadratic matrix equations, we study the square root of M-matrix, and prove that for a regular M-matrix there always exists a regular M-matrix as its square root. In addition, a structure-preserving doubling algorithm is proposed to compute the square root. Theoretical analysis and numerical experiments are given to show that our method is feasible and is effective under certain conditions.



    加载中


    [1] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, London, 1991.
    [2] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia London, 2008.
    [3] N. J. Higham, A. H. Al-Mohy, Computing matrix functions, Acta Numer., 19 (2010), 159–208. https://doi.org/10.1017/S0962492910000036 doi: 10.1017/S0962492910000036
    [4] D. A. Bini, B. Iannazzo, B. Meini, Numerical Solution of Algebraic Riccati Equations, SIAM, Philadelphia London, 2012.
    [5] G. Alefeld, N. Schneider, On square root of M-matrices, Linear Algebra Appl., 42 (1982), 119–132. https://doi.org/10.1016/0024-3795(82)90143-4 doi: 10.1016/0024-3795(82)90143-4
    [6] L. Lin, Z. Y. Liu, On the square root of an H-matrix with positive diagonal elements, Ann. Oper. Res., 103 (2001), 339–350. https://doi.org/10.1023/A:1012931928589 doi: 10.1023/A:1012931928589
    [7] C. R. Johnson, K. Okubo, R. Reams, Uniqueness of matrix square roots and an application, Linear Algebra Appl., 323 (2001), 51–60. https://doi.org/10.1016/S0024-3795(00)00243-3 doi: 10.1016/S0024-3795(00)00243-3
    [8] Z. Y. Liu, Y. L. Zhang, R. Ralha, Computing the square roots of matrices with central symmetry, Appl. Math. Comput., 186 (2007), 715–726. https://doi.org/10.1016/j.amc.2006.08.032 doi: 10.1016/j.amc.2006.08.032
    [9] J. R. Cardoso, C. S. Kenney, F. S. Leite, Computing the square root and logarithm of a real P-orthogonal matrix, Appl. Numer. Math., 46 (2003), 173–196. https://doi.org/10.1016/S0168-9274(03)00033-3 doi: 10.1016/S0168-9274(03)00033-3
    [10] C. B. Lu, C. Q. Gu, The computation of the square root of circulant matrices, Appl. Math. Comput., 217 (2011), 6819–6829. https://doi.org/10.1016/j.amc.2011.01.018 doi: 10.1016/j.amc.2011.01.018
    [11] Z. Y. Liu, Y. L. Zhang, J. Santos, R. Ralha, On computing complex square roots of real matrices, Appl. Math. Lett., 25 (2012), 1565–1568. https://doi.org/10.1016/j.aml.2012.01.015 doi: 10.1016/j.aml.2012.01.015
    [12] P. D. Moral, A. Niclas, A Taylor expansion of the square root matrix function, J. Math. Anal. Appl., 465 (2018), 259–2668. https://doi.org/10.1016/j.jmaa.2018.05.005 doi: 10.1016/j.jmaa.2018.05.005
    [13] Å. Björck, S. Hammarling, A Schur method for the square root of a matrix, Linear Algebra Appl., 52/53 (1983), 127–140. https://doi.org/10.1016/0024-3795(83)80010-X doi: 10.1016/0024-3795(83)80010-X
    [14] N. J. Higham, Computing real square roots of a real matrix, Linear Algebra Appl., 88-89 (1987), 405–430. https://doi.org/10.1016/0024-3795(87)90118-2 doi: 10.1016/0024-3795(87)90118-2
    [15] N. J. Higham, Newton's method for the matrix square root, Math. Comput., 46 (1986), 537–549. https://doi.org/10.2307/2007992 doi: 10.2307/2007992
    [16] N. J. Higham, Stable iterations for the matrix square root, Numer. Alg., 15 (1997), 227–242. https://doi.org/10.1023/A:1019150005407 doi: 10.1023/A:1019150005407
    [17] E. D. Denman, A. N. Beavers, The matrix sign function and computations in systems, Appl. Math. Comput., 2 (1976), 63–94. https://doi.org/10.1016/0096-3003(76)90020-5 doi: 10.1016/0096-3003(76)90020-5
    [18] M. A. Hasan, A power method for computing square roots of complex matrices, J. Math. Anal. Appl., 213 (1997), 393–405. https://doi.org/10.1006/jmaa.1997.5517 doi: 10.1006/jmaa.1997.5517
    [19] B. Meini, The matrix square root from a new functional perspective: Theoretical results and computational issues, SIAM J. Matrix Anal. Appl., 26 (2004), 362–376. https://doi.org/10.1137/S0895479803426656 doi: 10.1137/S0895479803426656
    [20] D. A. Bini, B. Meini, The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyonds, Numer. Algor., 51 (2009), 23–60. https://doi.org/10.1007/s11075-008-9253-0 doi: 10.1007/s11075-008-9253-0
    [21] A. Frommer, B. Hashemi, Verified computation of square roots of a matrix, SIAM J. Matrix Anal. Appl., 31 (2009), 1279–1302. https://doi.org/10.1137/090757058 doi: 10.1137/090757058
    [22] A. Sadeghi, Approximating the principal matrix square root using some novel third-order iterative methods, Ain Shams Eng. J., 9 (2018), 993–999. https://doi.org/10.1016/j.asej.2016.06.004 doi: 10.1016/j.asej.2016.06.004
    [23] C. Mo, D. Gerontitis, P. S. Stanimirović, Solving the time-varying tensor square root equation by varying-parameters finite-time Zhang neural network, Neurocomputing, 445 (2021), 309–325. https://doi.org/10.1016/j.neucom.2021.03.011 doi: 10.1016/j.neucom.2021.03.011
    [24] S. G. Evan, Zolotarev iterations for the matrix square root, SIAM J. Matrix Anal. Appl., 40 (2019), 696–719. https://doi.org/10.1137/18M1178529 doi: 10.1137/18M1178529
    [25] C. H. Guo, Explicit convergence regions of Newton's method and Chebyshev's method for the matrix pth root, Linear Algebra Appl., 583 (2019), 63–76. https://doi.org/10.1016/j.laa.2019.08.020 doi: 10.1016/j.laa.2019.08.020
    [26] S. Miyajima, Fast enclosure for a matrix inverse square root, Linear Algebra Appl., 467 (2015), 116–135. https://doi.org/10.1016/j.laa.2014.11.007 doi: 10.1016/j.laa.2014.11.007
    [27] X. F. Duan, C. Y. Wang, C. M. Li, Newton's method for solving the tensor square root problem, Appl. Math. Lett., 98 (2019), 57–62. https://doi.org/10.1016/j.aml.2019.05.031 doi: 10.1016/j.aml.2019.05.031
    [28] D. Lu, C. H. Guo, Explicit p-dependent convergence regions of Newton's method for the matrix pth root, Appl. Math. Lett., 122 (2021), 107566. https://doi.org/10.1016/j.aml.2021.107566 doi: 10.1016/j.aml.2021.107566
    [29] A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1994.
    [30] C. H. Guo, On algebraic Riccati equations associated with M-matrices, Linear Algebra Appl., 439 (2013), 2800–2814. https://doi.org/10.1016/j.laa.2013.08.018 doi: 10.1016/j.laa.2013.08.018
    [31] T. M. Huang, R. C. Li, W. W. Lin, Structure-Preserving Doubling Algorithms for Nonlinear Matrix Equations, SIAM, Philadelphia, 2018.
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Electronic Research Archive
1 1.3

Metrics

Article views(94) PDF downloads(11) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(4)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog