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Reasons for stability in the construction of derivative-free multistep iterative methods
  • Alicia Cordero,
  • B. Neta,
  • Juan Ramon Torregrosa
Alicia Cordero
Universitat Politecnica de Valencia

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B. Neta
Naval Postgraduate School
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Juan Ramon Torregrosa
Universitat Politecnica de Valencia
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In this paper, a deep dynamical analysis is made using tools from multidimensional real discrete dynamics of some derivative-free iterative methods with memory. They all have good qualitative properties, but one of them (due to Traub) shows the same behavior as Newton’s method on quadratic polynomials. Then, the same techniques are employed to analyze the performance of several multipoint schemes with memory, whose first step is Traub’s method, but their construction was made using different procedures. Therefore, their stability is analyzed, showing which is the best in terms of the wideness of basins of convergence or the existence of free critical points that would yield convergence towards different elements from the desired zeros of the nonlinear function. Therefore, the best stability properties are linked with the best estimations made in the iterative expressions rather than their simplicity. These results have been checked with a numerical and graphical comparison with many other known methods with and without memory, with different orders of convergence, with excellent performance.
22 Apr 2022Submitted to Mathematical Methods in the Applied Sciences
23 Apr 2022Submission Checks Completed
23 Apr 2022Assigned to Editor
24 May 2022Reviewer(s) Assigned
24 Nov 2022Review(s) Completed, Editorial Evaluation Pending
02 Dec 2022Editorial Decision: Revise Major
15 Dec 20221st Revision Received
16 Dec 2022Submission Checks Completed
16 Dec 2022Assigned to Editor
16 Dec 2022Review(s) Completed, Editorial Evaluation Pending
19 Dec 2022Reviewer(s) Assigned
17 Apr 2023Editorial Decision: Accept