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<p>In 2010, Soleymani introduced a three step iterative method which is of convergence order six. The method consists of a two step Jarratt like method followed by an extension to sixth order method using divided differences. The Jarratt like step in this method is of order four and shows very clear advantages over other similar order methods. But the results are presented in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>. To prove the Jarratt-like method is of order four, the Taylor series is used, and to proceed with this analysis, one needs to assume that the function is at least five times differentiable. In this scenario, we give another proof which uses only assumptions on the first three derivatives, and the results are presented in a general setting such that the function is defined on a Banach space. We also present two extensions of the method which are of order six. Comparison studies have been done with other fourth order methods and sixth order methods. To check the stability of the methods, the dynamic results for some examples are also presented.</p>

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Convergence order of efficient Jarratt-like method with relaxed differentiability assumptions

  • 期刊论文

  • 发表日期：

    2026年1月8日

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  • Ajil Kunnarath,
  • Manjusree Gopal,
  • Santhosh George,
  • P. Jidesh,
  • Ioannis K. Argyros

摘要

In 2010, Soleymani introduced a three step iterative method which is of convergence order six. The method consists of a two step Jarratt like method followed by an extension to sixth order method using divided differences. The Jarratt like step in this method is of order four and shows very clear advantages over other similar order methods. But the results are presented in \(\mathbb {R}\) R . To prove the Jarratt-like method is of order four, the Taylor series is used, and to proceed with this analysis, one needs to assume that the function is at least five times differentiable. In this scenario, we give another proof which uses only assumptions on the first three derivatives, and the results are presented in a general setting such that the function is defined on a Banach space. We also present two extensions of the method which are of order six. Comparison studies have been done with other fourth order methods and sixth order methods. To check the stability of the methods, the dynamic results for some examples are also presented.