Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations

Ioannis K. Argyros () and Santhosh George
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Ioannis K. Argyros: Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
Santhosh George: Department of Mathematical & Computational Science, National Institute of Technology Karnataka, Surathkal, Mangaluru 575 025, India

Mathematics, 2024, vol. 13, issue 1, 1-19

Abstract: Chebyshev-type methods have replaced the Chebyshev method in practice for solving nonlinear equations in abstract spaces. These methods are of the same R-order of three. However, they are easier to deal with, since the computationally expensive second derivative of the operator involved does not appear on these methods. However, the invertibility of the first derivative is still required at each step of the iteration. In this article, the inverse is replaced by a finite sum of linear operators. The convergence of the new Hybrid Chebyshev-Type Method (HCTM) is established under relaxed generalized continuity assumptions on the derivative and majorizing sequences. The iterates of the new methods converge to the original ones, but they are easier to find. Moreover, the numerical examples demonstrate that the new iterates converge essentially as fast to the solution. The methodology of this article can be used on other methods with inverses along the same lines due to its generality.

Keywords: Chebyshev method; optimized and hybrid Chebyshev-type methods; Banach space; convergence; inverse of an operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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