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<p>This study presents a Mean Value Theorem (MVT) based convergence analysis for a highly efficient iterative algorithm of order eight studied by Cordero et al. (2018) for solving nonlinear systems of equations. Using the same idea, one can increase the order of convergence (OC) from <i>p</i> to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p+3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>+</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> for any algorithm with OC <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p\ge 5.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>5</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The algorithm involves a weight function, so one can introduce new algorithms by choosing the weight function appropriately. We have proved the OC using MVT and in a Banach space setting. Both local and semi-local analysis are discussed under the same set of assumptions. Our study used the derivative of <i>T</i> only up to the third order. This algorithm can be applied to solve complex problems from various scientific disciplines. Algorithm’s performance is illustrated using numerical examples.</p>

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A unified semi-local and local analysis of a highly efficient iterative algorithms for nonlinear equations

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Santhosh George,
  • Manjusree Gopal,
  • Samhitha Bhide

摘要

This study presents a Mean Value Theorem (MVT) based convergence analysis for a highly efficient iterative algorithm of order eight studied by Cordero et al. (2018) for solving nonlinear systems of equations. Using the same idea, one can increase the order of convergence (OC) from p to \(p+3\) p + 3 for any algorithm with OC \(p\ge 5.\) p ≥ 5 . The algorithm involves a weight function, so one can introduce new algorithms by choosing the weight function appropriately. We have proved the OC using MVT and in a Banach space setting. Both local and semi-local analysis are discussed under the same set of assumptions. Our study used the derivative of T only up to the third order. This algorithm can be applied to solve complex problems from various scientific disciplines. Algorithm’s performance is illustrated using numerical examples.