Notes on iterative processes

Abstract

If ηn is the error in the result of repeating an iterative process n times, and ηn+1 is the iterative process is called Kth order. It is shown that if for a given equation there is an iterative process of the (K + 1)th order, the iterative process of the Kth order is not unique, and conversely if an iterative process of the Kth order is not unique, it is generally possible to construct from two of the Kth order processes a process of the (K + 1)th order. As an example, three second-order processes for a square root are exhibited, and a third-order process is derived from two of them.Iterative processes for positive and negative integer roots are given, of kinds suitable for use on machines in which division is a relatively slow process and one to be used sparingly. It is shown how a second-order process can be derived from the results of two repetitions of a first-order process. The extension of this iterative process for the solution of differential and integral equations is a development which is urgently required.