A quartically fast iteration solver with convergence analysis for numerically determining the sign of a matrix

Ying Liu; Runqi Xue; Tao Liu; Shuai Wang; Stanford Shateyi; Ying Liu; Runqi Xue; Tao Liu; Shuai Wang; Stanford Shateyi

doi:10.3934/math.2025632

AIMS Mathematics

2025, Volume 10, Issue 6: 14055-14070. doi: 10.3934/math.2025632

Previous Article Next Article

Research article Special Issues

A quartically fast iteration solver with convergence analysis for numerically determining the sign of a matrix

1.
Foundation Department, Changchun Guanghua University, Changchun 130033, China
2.
School of Mathematics, Sichuan University, Chengdu 610041, China
3.
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
4.
Foundation Department, Changchun Guanghua University, Changchun 130033, China
5.
Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa

Received: 12 February 2025 Revised: 16 April 2025 Accepted: 07 May 2025 Published: 18 June 2025
MSC : 41A25, 65F60

The calculation of the matrix sign function (MSF) is pivotal in numerous mathematical contexts, offering a matrix-based transformation that identifies the sign for every eigenvalue within an invertible matrix. This paper introduces a new iteration procedure tailored to effectively compute the MSF, with a particular focus on expanding the order of convergence. Our proposed solver achieves fourth-order convergence, rendering it effective for a broad spectrum of matrices. Numerical experiments for both real and complex matrices are included to support the derivations.
- matrix sign,
- convergence,
- initial matrix,
- numerical scheme,
- Newton's solver
Citation: Ying Liu, Runqi Xue, Tao Liu, Shuai Wang, Stanford Shateyi. A quartically fast iteration solver with convergence analysis for numerically determining the sign of a matrix[J]. AIMS Mathematics, 2025, 10(6): 14055-14070. doi: 10.3934/math.2025632

Related Papers:

Abstract

The calculation of the matrix sign function (MSF) is pivotal in numerous mathematical contexts, offering a matrix-based transformation that identifies the sign for every eigenvalue within an invertible matrix. This paper introduces a new iteration procedure tailored to effectively compute the MSF, with a particular focus on expanding the order of convergence. Our proposed solver achieves fourth-order convergence, rendering it effective for a broad spectrum of matrices. Numerical experiments for both real and complex matrices are included to support the derivations.

References

[1]	L. Hogben, Handbook of linear algebra, 2 Eds., New York: Chapman and Hall/CRC, 2013. https://doi.org/10.1201/b16113
[2]	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
[3]	M. Kansal, V. Sharma, P. Sharma, L. Jäntschi, A globally convergent iterative method for matrix sign function and its application for determining the eigenvalues of a matrix pencil, Symmetry, 16 (2024), 481. https://doi.org/10.3390/sym16040481 doi: 10.3390/sym16040481
[4]	S. Wang, Z. Wang, W. Xie, Y. Qi, T. Liu, An accelerated sixth-order procedure to determine the matrix sign function computationally, Mathematics, 13 (2025), 1080. https://doi.org/10.3390/math13071080 doi: 10.3390/math13071080
[5]	J. D. Roberts, Linear model reduction and solution of the algebraic Riccati equation by use of the sign function, Int. J. Control, 32 (1980), 677–687. https://doi.org/10.1080/00207178008922881 doi: 10.1080/00207178008922881
[6]	N. J. Higham, Functions of matrices: Theory and computation, Philadelphia: Society for Industrial and Applied Mathematics, 2008. https://doi.org/10.1137/1.9780898717778
[7]	A. R. Soheili, M. Amini, F. Soleymani, A family of Chaplygin–type solvers for Itô stochastic differential equations, Appl. Math. Comput., 340 (2019), 296–304. https://doi.org/10.1016/j.amc.2018.08.038 doi: 10.1016/j.amc.2018.08.038
[8]	A. R. Soheili, F. Toutounian, F. Soleymani, A fast convergent numerical method for matrix sign function with application in SDEs, J. Comput. Appl. Math., 282 (2015), 167–178. https://doi.org/10.1016/j.cam.2014.12.041 doi: 10.1016/j.cam.2014.12.041
[9]	J. Golzarpoor, D. Ahmed, S. Shateyi, Constructing a matrix mid-point iterative method for matrix square roots and applications, Mathematics, 10 (2022), 2200. https://doi.org/10.3390/math10132200 doi: 10.3390/math10132200
[10]	J. Saak, S. W. R. Werner, Using $L D L^T$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations, Electron. Trans. Numer. Anal., 62 (2024), 95–118. https://doi.org/10.1553/etna_vol62s95 doi: 10.1553/etna_vol62s95
[11]	C. S. Kenney, A. J. Laub, Rational iterative methods for the matrix sign function, SIAM J. Matrix Anal. Appl., 12 (1991), 273–291. https://doi.org/10.1137/0612020 doi: 10.1137/0612020
[12]	D. Jung, C. Chun, A general approach for improving the Padé iterations for the matrix sign function, J. Comput. Appl. Math., 436 (2024), 115348. https://doi.org/10.1016/j.cam.2023.115348 doi: 10.1016/j.cam.2023.115348
[13]	F. Soleymani, P. S. Stanimirović, S. Shateyi, F. K. Haghani, Approximating the matrix sign function using a novel iterative method, Abstr. Appl. Anal., 2014 (2014), 105301. https://doi.org/10.1155/2014/105301 doi: 10.1155/2014/105301
[14]	O. Gomilko, F. Greco, K. Ziȩtak, A Padé family of iterations for the matrix sign function and related problems, Numer. Linear Algebra Appl., 19 (2012), 585–605. https://doi.org/10.1002/nla.786 doi: 10.1002/nla.786
[15]	A. Cordero, F. Soleymani, J. R. Torregrosa, M. Zaka Ullah, Numerically stable improved Chebyshev–Halley type schemes for matrix sign function, J. Comput. Appl. Math., 318 (2017), 189–198. https://doi.org/10.1016/j.cam.2016.10.025 doi: 10.1016/j.cam.2016.10.025
[16]	J. F. Traub, Iterative methods for the solution of equations, New York: Prentice-Hall, 1964.
[17]	M. Z. Ullah, S. M. Alaslani, F. O. Mallawi, F. Ahmad, S. Shateyi, M. Asma, A fast and efficient Newton-type iterative scheme to find the sign of a matrix, AIMS Mathematics, 8 (2023), 19264–19274. https://doi.org/10.3934/math.2023982 doi: 10.3934/math.2023982
[18]	F. Soleymani, A. Kumar, A fourth-order method for computing the sign function of a matrix with application in the Yang–Baxter-like matrix equation, Comput. Appl. Math., 38 (2019), 64. https://doi.org/10.1007/s40314-019-0816-6 doi: 10.1007/s40314-019-0816-6
[19]	R. Bhatia, Matrix analysis, New York: Springer, 1997. https://doi.org/10.1007/978-1-4612-0653-8
[20]	L. Shi, M. Z. Ullah, H. K. Nashine, M. Alansari, S. Shateyi, An enhanced numerical iterative method for expanding the attraction basins when computing matrix signs of invertible matrices, Fractal Fract., 7 (2023), 684. https://doi.org/10.3390/fractalfract7090684 doi: 10.3390/fractalfract7090684
[21]	Y. Feng, A. Z. Othman, An accelerated iterative method to find the sign of a nonsingular matrix with quartical convergence, Iran. J. Sci., 47 (2023), 1359–1366. https://doi.org/10.1007/s40995-023-01506-7 doi: 10.1007/s40995-023-01506-7
[22]	B. Iannazzo, Numerical solution of certain nonlinear matrix equations, PhD Thesis, Universita Degli Studi di Pisa, Pisa, Italy, 2007.
[23]	G. W. Stewart, Introduction to matrix computations, New York: Academic Press, 1973.
[24]	S. Mangano, Mathematica cookbook, O'Reilly Media, 2010.
[25]	H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, New York: Springer, 2011. https://doi.org/10.1007/978-0-387-70914-7
[26]	K. Yosida, Functional analysis, Berlin, Heidelberg: Springer, 1995. https://doi.org/10.1007/978-3-642-61859-8
[27]	W. Rudin, Functional analysis, McGraw Hill, 1991.

Reader Comments

Your name:*

Email:*
© 2025 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)