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A Study of at Least Sixth Convergence Order Methods Without or with Memory and Divided Differences for Equations Under Generalized Continuity

Author

Listed:
  • Ioannis K. Argyros

    (Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA)

  • Ramandeep Behl

    (Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
    Department of Mathematics, Saveetha School of Engineering, SIMATS, Chennai 602105, India)

  • Sattam Alharbi

    (Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia)

  • Abdulaziz Mutlaq Alotaibi

    (Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia)

Abstract

Multistep methods typically use Taylor series to attain their convergence order, which necessitates the existence of derivatives not naturally present in the iterative functions. Other issues are the absence of a priori error estimates, information about the radius of convergence or the uniqueness of the solution. These restrictions impose constraints on the use of such methods, especially since these methods may converge. Consequently, local convergence analysis emerges as a more effective approach, as it relies on criteria involving only the operators of the methods. This expands the applicability of such methods, including in non-Euclidean space scenarios. Furthermore, this work uses majorizing sequences to address the more challenging semi-local convergence analysis, which was not explored in earlier research. We adopted generalized continuity constraints to control the derivatives and obtain sharper error estimates. The sufficient convergence criteria are demonstrated through examples.

Suggested Citation

  • Ioannis K. Argyros & Ramandeep Behl & Sattam Alharbi & Abdulaziz Mutlaq Alotaibi, 2025. "A Study of at Least Sixth Convergence Order Methods Without or with Memory and Divided Differences for Equations Under Generalized Continuity," Mathematics, MDPI, vol. 13(5), pages 1-23, February.
  • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:799-:d:1601761
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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. Cordero, Alicia & Rojas-Hiciano, Renso V. & Torregrosa, Juan R. & Vassileva, Maria P., 2025. "Maximally efficient damped composed Newton-type methods to solve nonlinear systems of equations," Applied Mathematics and Computation, Elsevier, vol. 492(C).
    3. Alicia Cordero & Eva G. Villalba & Juan R. Torregrosa & Paula Triguero-Navarro, 2021. "Convergence and Stability of a Parametric Class of Iterative Schemes for Solving Nonlinear Systems," Mathematics, MDPI, vol. 9(1), pages 1-18, January.
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