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

Jiameihui Zhu, Yutong Li, Yilin Li, Tao Liu () and Qiang Ma
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Jiameihui Zhu: School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
Yutong Li: School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
Yilin Li: School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
Tao Liu: School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
Qiang Ma: Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China

Mathematics, 2025, vol. 13, issue 21, 1-18

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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