In this paper, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations assuming that the first-order Fr\'echet derivative belongs to the Lipschitz class. The significance of our work is that it avoids the standard practice of Taylor expansion thereby, extends the applicability of the scheme by applying the technique based on the first-order derivative only. Also, this study provides radii of balls of convergence, the error bounds in terms of distances in addition to the uniqueness of the solution. Furthermore, generalization of this analysis satisfying H\"{o}lder continuity condition is provided since it is more relaxed than Lipschitz continuity condition. We have considered some numerical examples and computed the radii of the convergence balls.

翻译：本文介绍了对七级多步法的本地趋同分析,用于解决非线性方程,假定Fr\'echet衍生物属于Lipschitz类,我们工作的意义在于它避免了泰勒扩展的标准做法,从而通过只应用以一阶衍生物为基础的技术,扩大了该计划的适用性。此外,本研究报告提供了趋同球的半径,差幅除了这一解决办法的独特性之外还从距离的角度来算。此外,还提供了满足H\"{o}lderer连续性条件的这一分析的概括性,因为它比Lipschitz的连续性条件更加宽松。我们考虑了一些数字例子并计算了趋同球的半径。