Many problems in applied mathematics require root finding algorithms. Unfortunately, root finding methods have limitations. Firstly, regarding the convergence, there is a trade-off between the size of it's domain and it's rate. Secondly the numerous evaluations of the function and its derivatives penalize the efficiency of high order methods. In this article, we present a family of high order methods, that require few functional evaluations ( One for each step plus one for each considered derivative at the start of the method), thus increasing the efficiency of the methods.

翻译：应用数学中的许多问题需要求根算法。然而，求根方法存在局限性。首先，就收敛性而言，其收敛域的大小与收敛速度之间存在权衡。其次，对函数及其导数的大量求值限制了高阶方法的效率。在本文中，我们提出了一系列高阶方法，该方法只需少量函数求值（每一步一次求值，加上方法开始时对每个考虑导数的一次求值），从而提高了方法的效率。