Using Matrix Eigenvalues to Construct an Iterative Method with the Highest Possible Efficiency Index Two

Malik Zaka Ullah, Vali Torkashvand, Stanford Shateyi and Mir Asma
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Malik Zaka Ullah: Mathematical Modeling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Vali Torkashvand: Member of Young Researchers and Elite Club, Shahr-e-Qods Branch, Islamic Azad University, Tehran 37515-374, Iran
Stanford Shateyi: Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa
Mir Asma: Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia

Mathematics, 2022, vol. 10, issue 9, 1-15

Abstract: In this paper, we first derive a family of iterative schemes with fourth order. A weight function is used to maintain its optimality. Then, we transform it into methods with several self-accelerating parameters to reach the highest possible convergence rate 8. For this aim, we employ the property of the eigenvalues of the matrices and the technique with memory. Solving several nonlinear test equations shows that the proposed variants have a computational efficiency index of two (maximum amount possible) in practice.

Keywords: with-memory method; accelerator parameter; R -order convergence; eigenvalues (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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