A fast and efficient Newton-type iterative scheme to find the sign of a matrix

Malik Zaka Ullah; Sultan Muaysh Alaslani; Fouad Othman Mallawi; Fayyaz Ahmad; Stanford Shateyi; Mir Asma; Malik Zaka Ullah; Sultan Muaysh Alaslani; Fouad Othman Mallawi; Fayyaz Ahmad; Stanford Shateyi; Mir Asma

doi:10.3934/math.2023982

AIMS Mathematics

2023, Volume 8, Issue 8: 19264-19274. doi: 10.3934/math.2023982

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Research article

A fast and efficient Newton-type iterative scheme to find the sign of a matrix

1.
Mathematical Modeling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, School of Science, National Textile University, Faisalabad, Pakistan
3.
Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa
4.
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia

Received: 11 July 2022 Revised: 19 October 2022 Accepted: 07 November 2022 Published: 07 June 2023
MSC : 65F30, 65F60

This work proposes a new scheme under the umbrella of iteration methods to compute the sign of an invertible matrix. To this target, a review of the exiting solvers of the same type is given and then a new scheme is derived based on a multi-step Newton-type nonlinear equation solver. It is shown that the new method and its reciprocal converge globally with wider convergence radii in contrast to their competitors of the same order from the general Padé schemes. After investigation on the theoretical parts, numerical experiments based on complex matrices of various sizes are furnished to reveal the superiority of the proposed solver in terms of elapsed CPU time.
- matrix sign function,
- iterative methods,
- fourth order,
- stability,
- basins of attraction
Citation: Malik Zaka Ullah, Sultan Muaysh Alaslani, Fouad Othman Mallawi, Fayyaz Ahmad, Stanford Shateyi, Mir Asma. A fast and efficient Newton-type iterative scheme to find the sign of a matrix[J]. AIMS Mathematics, 2023, 8(8): 19264-19274. doi: 10.3934/math.2023982

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Abstract

This work proposes a new scheme under the umbrella of iteration methods to compute the sign of an invertible matrix. To this target, a review of the exiting solvers of the same type is given and then a new scheme is derived based on a multi-step Newton-type nonlinear equation solver. It is shown that the new method and its reciprocal converge globally with wider convergence radii in contrast to their competitors of the same order from the general Padé schemes. After investigation on the theoretical parts, numerical experiments based on complex matrices of various sizes are furnished to reveal the superiority of the proposed solver in terms of elapsed CPU time.

References

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