Extending the Applicability of a Two-Step Vectorial Method with Accelerators of Order Five for Solving Systems of Equations
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 t...
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2025-04-01
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| author | Ioannis K. Argyros Stepan Shakhno Yurii Shunkin Samundra Regmi Nirjal Shrestha |
| author_facet | Ioannis K. Argyros Stepan Shakhno Yurii Shunkin Samundra Regmi Nirjal Shrestha |
| author_sort | Ioannis K. Argyros |
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| description | 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. |
| format | Article |
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| issn | 2227-7390 |
| language | English |
| publishDate | 2025-04-01 |
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| series | Mathematics |
| spelling | doaj-art-fab6890343ba4b0da4e47e86b40503e32025-08-20T02:28:32ZengMDPI AGMathematics2227-73902025-04-01138129910.3390/math13081299Extending the Applicability of a Two-Step Vectorial Method with Accelerators of Order Five for Solving Systems of EquationsIoannis K. Argyros0Stepan Shakhno1Yurii Shunkin2Samundra Regmi3Nirjal Shrestha4Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USADepartment of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, UkraineDepartment of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, UkraineDepartment of Mathematics, University of Houston, Houston, TX 77205, USADepartment of Mathematics, University of Florida, Gainesville, FL 32603, USAThe 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.https://www.mdpi.com/2227-7390/13/8/1299two-step vectorial method with acceleratorsconvergenceorder five |
| spellingShingle | Ioannis K. Argyros Stepan Shakhno Yurii Shunkin Samundra Regmi Nirjal Shrestha Extending the Applicability of a Two-Step Vectorial Method with Accelerators of Order Five for Solving Systems of Equations Mathematics two-step vectorial method with accelerators convergence order five |
| title | Extending the Applicability of a Two-Step Vectorial Method with Accelerators of Order Five for Solving Systems of Equations |
| title_full | Extending the Applicability of a Two-Step Vectorial Method with Accelerators of Order Five for Solving Systems of Equations |
| title_fullStr | Extending the Applicability of a Two-Step Vectorial Method with Accelerators of Order Five for Solving Systems of Equations |
| title_full_unstemmed | Extending the Applicability of a Two-Step Vectorial Method with Accelerators of Order Five for Solving Systems of Equations |
| title_short | Extending the Applicability of a Two-Step Vectorial Method with Accelerators of Order Five for Solving Systems of Equations |
| title_sort | extending the applicability of a two step vectorial method with accelerators of order five for solving systems of equations |
| topic | two-step vectorial method with accelerators convergence order five |
| url | https://www.mdpi.com/2227-7390/13/8/1299 |
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