The purpose of this paper is to suggest an approach for increasing the convergence speed of Halley’s method to solve a non-linear equation. This approach is based on the second order Taylor polynomial and on Halley’s formula. By applying it a certain number of times, we obtain a new family of methods. The originality of this family is manifested in the fact that all its sequences are generated from one exceptional formula that depends on a natural integer parameter $p$. In addition, under certain conditions, the convergence speed of its sequences increases with $p$. The convergence analysis shows that the order of convergence of all proposed methods is three. A study on their global convergence is carried out. To illustrate the performance of this family, several numerical comparisons are made with other third and higher order methods.

本文的目的是提出一种提高哈雷方法求解非线性方程的收敛速度的方法。该方法基于二阶泰勒多项式和哈雷公式。通过多次应用，我们获得了新的方法系列。该家族的独创性体现在以下事实：其所有序列均由一个依赖于自然整数参数的特殊公式生成$p$。另外，在一定条件下，其序列的收敛速度随着$p$。收敛分析表明，所有提出的方法的收敛顺序为三。对它们的全球趋同进行了研究。为了说明该系列的性能，使用其他三阶和更高阶方法进行了一些数值比较。