Optimal Starting Approximations for Newton's Method

Abstract

Various writers have dealt with the subject of optimal starting approxi- mations for square-root calculation by Newton's method. Three optimality criteria that have been used can be shown to lead to closely related approximations. This fact makes it surprisingly easy to choose a starting approximation of some pre- scribed form so that the maximum relative error after any number of Newton iterations is as small as possible. U 1. Introduction. The choice of polynomial and rational starting approximations for square-root calculation by Newton's method has been the subject of various investigations (e.g., (1)-(5)). These approach the problem from several different points of view. In this paper we will show how approximations obtained from these different viewpoints are related and how some of them can be derived from others. The problem of evaluating V/x for any x > 0 is easily reduced to the problem of evaluating V/x for x in some closed interval (a, b) such that 0 < a < b. Here (a, b) depends on the radix of the floating-point number system of the computer to be used; typical possibilities are (1/16, 1) and (-, 2). The following procedure is used to compute an approximate value for VIx. Using a polynomial or rational approxima- tion f(x) to V/x, valid in (a, b), compute a starting value yo = f(x) and then obtain Y1, Y22 .2 Yn by means of the relation Yk+1 = 12 (Yk + Xl8k), k = O. 1, * ,n - 1 . Then Yn V/x. It is customary not to test for convergence, since the number of iterations required in practice is quite small. Instead, the number of iterations n is