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Some higher-order iteration functions for solving nonlinear models

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  • Alzahrani, Abdullah Khamis Hassan
  • Behl, Ramandeep
  • Alshomrani, Ali Saleh

Abstract

In this paper, we present a new efficient sixth-order family of Jarratt type methods for solving scalar equations. Then, we extend this family to the multidimensional case preserving the same order of convergence. We also discuss the theoretical convergence properties of the proposed scheme in the case of scalar as well as multidimensional case. The derivation of these schemes are based on weight function approach and free disposable parameters. We also demonstrate the applicability of them on total six number of problems: first five are real life problems namely, continuous stirred tank reactor (CSTR), chemical engineering, the trajectory of an electron in the air gap between two parallel plates, Hammerstein integration and boundary value problems; last one is the standard academic test problem. In addition, numerical comparisons are made to show the performance of the proposed iterative techniques with the existing techniques of the same order in the scalar as well as multi-dimensional case. Finally on the basis of numerical results, we conclude that our techniques perform better in terms of residual error, error between the two consecutive iterations, asymptotic error constant term and approximated root as compared to the existing ones of same order in scalar as well as multidimensional case.

Suggested Citation

  • Alzahrani, Abdullah Khamis Hassan & Behl, Ramandeep & Alshomrani, Ali Saleh, 2018. "Some higher-order iteration functions for solving nonlinear models," Applied Mathematics and Computation, Elsevier, vol. 334(C), pages 80-93.
    • Handle: RePEc:eee:apmaco:v:334:y:2018:i:c:p:80-93
    DOI: 10.1016/j.amc.2018.03.120
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    References listed on IDEAS

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    1. Artidiello, Santiago & Cordero, Alicia & Torregrosa, Juan R. & Vassileva, Maria P., 2015. "Multidimensional generalization of iterative methods for solving nonlinear problems by means of weight-function procedure," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 1064-1071.
    2. Lee, Min-Young & Ik Kim, Young & Alberto Magreñán, Á., 2017. "On the dynamics of a triparametric family of optimal fourth-order multiple-zero finders with a weight function of the principal mth root of a function-to function ratio," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 564-590.
    3. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2015. "On developing a higher-order family of double-Newton methods with a bivariate weighting function," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 277-290.
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    Cited by:

    1. Deepak Kumar & Ioannis K. Argyros & Janak Raj Sharma, 2018. "Convergence Ball and Complex Geometry of an Iteration Function of Higher Order," Mathematics, MDPI, vol. 7(1), pages 1-13, December.
    2. Hessah Faihan Alqahtani & Ramandeep Behl & Munish Kansal, 2019. "Higher-Order Iteration Schemes for Solving Nonlinear Systems of Equations," Mathematics, MDPI, vol. 7(10), pages 1-14, October.
    3. José J. Padilla & Francisco I. Chicharro & Alicia Cordero & Alejandro M. Hernández-Díaz & Juan R. Torregrosa, 2024. "A Class of Efficient Sixth-Order Iterative Methods for Solving the Nonlinear Shear Model of a Reinforced Concrete Beam," Mathematics, MDPI, vol. 12(3), pages 1-16, February.
    4. Janak Raj Sharma & Deepak Kumar & Ioannis K. Argyros, 2019. "Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations," Mathematics, MDPI, vol. 7(12), pages 1-16, December.

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