A Rapidly Convergent Iteration Scheme for Computing the Matrix Sign Function of Invertible Matrices

Articles in Press

Document Type : Research Paper

Authors

1 Mathematical Modelling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia.

2 Department of Mathematics, Jamoum University College, Umm Al-Qura University, Makkah, 25375, Saudi Arabia.

3 Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, Tamil Nadu, India.

10.22034/cmde.2025.67379.3203

Abstract

This paper presents a novel 6th-rate multi-step iteration method to calculate the matrix sign function of an invertible matrix. The proposed scheme is constructed using rational approximations and carefully designed weight functions, leading to enhanced computational efficiency and high accuracy. A detailed convergence analysis is provided, including the derivation of an explicit error expression that rigorously establishes the method's sixth-order convergence rate. Additionally, we generalize the approach to matrix iteration frameworks, demonstrating that the eigenvalues of the generated iterates asymptotically converge to their theoretical limits. Numerical tests are conducted to validate the analytical findings and illustrate the superior performance of the presented method compared to existing techniques.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 12 April 2026

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History

  • Receive Date: 18 May 2025
  • Revise Date: 08 October 2025
  • Accept Date: 12 April 2026

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