Notes on iterative processes

If η n is the error in the result of repeating an iterative process n times, and η n +1 is the iterative process is called K th 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 K th order is not unique, and conversely if an iterative process of the K th order is not unique, it is generally possible to construct from two of the K th 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.