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Volume 13
Issue 21
10.3390/math13213370
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Article

Fourth-Order Iterative Algorithms for the Simultaneous Calculation of Matrix Square Roots and Their Inverses

by
Jiameihui Zhu
1,
Yutong Li
1,
Yilin Li
1,
Tao Liu
1,* and
Qiang Ma
2
1
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
2
Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(21), 3370; https://doi.org/10.3390/math13213370
Submission received: 27 September 2025 / Revised: 17 October 2025 / Accepted: 21 October 2025 / Published: 22 October 2025
(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis: 2nd Edition)
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Abstract

This paper develops and analyzes new high-order iterative schemes for the effective evaluation of the matrix square root (MSR). By leveraging connections between the matrix sign function and the MSR, we design stable algorithms that exhibit fourth-order convergence under mild spectral conditions. Detailed error bounds and convergence analyses are provided, ensuring both theoretical rigor and numerical reliability. A comprehensive set of numerical experiments, conducted across structured and large-scale test matrices, demonstrates the superior performance of the proposed methods compared to classical approaches, both in terms of computational efficiency and accuracy. The results confirm that the proposed iterative strategies provide robust and scalable tools for practical applications requiring repeated computation of matrix square roots.
Keywords: matrix square root; iterative methods; convergence analysis; computational efficiency; Jordan canonical form matrix square root; iterative methods; convergence analysis; computational efficiency; Jordan canonical form

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MDPI and ACS Style

Zhu, J.; Li, Y.; Li, Y.; Liu, T.; Ma, Q. Fourth-Order Iterative Algorithms for the Simultaneous Calculation of Matrix Square Roots and Their Inverses. Mathematics 2025, 13, 3370. https://doi.org/10.3390/math13213370

AMA Style

Zhu J, Li Y, Li Y, Liu T, Ma Q. Fourth-Order Iterative Algorithms for the Simultaneous Calculation of Matrix Square Roots and Their Inverses. Mathematics. 2025; 13(21):3370. https://doi.org/10.3390/math13213370

Chicago/Turabian Style

Zhu, Jiameihui, Yutong Li, Yilin Li, Tao Liu, and Qiang Ma. 2025. "Fourth-Order Iterative Algorithms for the Simultaneous Calculation of Matrix Square Roots and Their Inverses" Mathematics 13, no. 21: 3370. https://doi.org/10.3390/math13213370

APA Style

Zhu, J., Li, Y., Li, Y., Liu, T., & Ma, Q. (2025). Fourth-Order Iterative Algorithms for the Simultaneous Calculation of Matrix Square Roots and Their Inverses. Mathematics, 13(21), 3370. https://doi.org/10.3390/math13213370

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