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Extending the Applicability of a Two-Step Vectorial Method with Accelerators of Order Five for Solving Systems of Equations

Author

Listed:
  • Ioannis K. Argyros

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

  • Stepan Shakhno

    (Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine)

  • Yurii Shunkin

    (Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine)

  • Samundra Regmi

    (Department of Mathematics, University of Houston, Houston, TX 77205, USA)

  • Nirjal Shrestha

    (Department of Mathematics, University of Florida, Gainesville, FL 32603, USA)

Abstract

The local convergence analysis of a two-step vectorial method of accelerators with order five has been shown previously. But the convergence order five is obtained using Taylor series and assumptions on the existence of at least the fifth derivative of the mapping involved, which is not present in the method. These assumptions limit the applicability of the method. Moreover, a priori error estimates or the radius of convergence or uniqueness of the solution results have not been given. All these concerns are addressed in this paper. Furthermore, the more challenging semi-local convergence analysis, not previously studied, is presented using majorizing sequences. The convergence for both analyses depends on the generalized continuity of the Jacobian of the mapping involved, which is used to control it and sharpen the error distances. Numerical examples validate the sufficient convergence conditions presented in the theory.

Suggested Citation

  • Ioannis K. Argyros & Stepan Shakhno & Yurii Shunkin & Samundra Regmi & Nirjal Shrestha, 2025. "Extending the Applicability of a Two-Step Vectorial Method with Accelerators of Order Five for Solving Systems of Equations," Mathematics, MDPI, vol. 13(8), pages 1-19, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:8:p:1299-:d:1635352
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    References listed on IDEAS

    as
    1. Abbasbandy, Saeid & Bakhtiari, Parisa & Cordero, Alicia & Torregrosa, Juan R. & Lotfi, Taher, 2016. "New efficient methods for solving nonlinear systems of equations with arbitrary even order," Applied Mathematics and Computation, Elsevier, vol. 287, pages 94-103.
    2. Budzko, Dzmitry & Cordero, Alicia & Torregrosa, Juan R., 2015. "A new family of iterative methods widening areas of convergence," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 405-417.
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