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Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations

Author

Listed:
  • 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)

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.

Suggested Citation

  • Ioannis K. Argyros & Santhosh George, 2024. "Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations," Mathematics, MDPI, vol. 13(1), pages 1-19, December.
    • Handle: RePEc:gam:jmathe:v:13:y:2024:i:1:p:74-:d:1555424
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    References listed on IDEAS

    as
    1. Ioannis K. Argyros & Santhosh George & Stepan Shakhno & Samundra Regmi & Mykhailo Havdiak & Michael I. Argyros, 2024. "Asymptotically Newton-Type Methods without Inverses for Solving Equations," Mathematics, MDPI, vol. 12(7), pages 1-19, April.
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