In this article, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations. The point worth noting in our paper is that our analysis requires a weak hypothesis where the Fr\'echet derivative of the nonlinear operator satisfies the $\psi$-continuity condition and extends the applicability of the computation when both Lipschitz and H\"{o}lder conditions fail. The convergence in this study is shown under the hypotheses on the first order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some numerical examples are provided to show the performance of the method presented in this contribution over some existing schemes.
翻译:在本篇文章中,为解决非线性方程,介绍了对多步第七级方法的本地趋同分析。值得指出的是,我们的分析需要一个薄弱的假设,即非线性经营人的Fr\'echet衍生物符合$\psi$-continity条件,并在Lipschitz和H\"{o}lder条件均不合格时扩大计算的适用性。本研究的趋同情况见于关于第一级衍生物的假设中,而不涉及较高级的衍生物。为了找到最初趋同领域的一部分,我们设计了一个战略。结果,新的Lipschitz常数至少与旧常数一样紧,从而使得对本地趋同情况进行更精确的趋同分析。提供了一些数字例子,以显示该方法对一些现有方案的表现。