The authors address the problem of finding optimal starting values in terms of absolute error for Newton’s method when it is applied to finding square roots. This absolute error approach, as compared to the more extensively studied relative error method, has the advantage that it is known how many digits are significant and, in addition, this number is constant over the interval in which the computation is carried out. The authors provide a detailed analysis of the two approaches and tables of results for comparison. Three techniques exist to determine starting values. According to the authors, the first two are (1) optimal initialization, in which a function of a given class is defined that provides the starting value for every x in the interval [a,b]; and (2) the table method, in which the interval [a,b] is divided into sub-intervals and different starting values are associated with each sub-interval. The third technique is a hybrid of the above two methods.

A good introduction explains the motivation for this work, which is welcome, since it is not immediately apparent from the title. A comprehensive list of references appears at the end of the paper. The paper will be of interest to all those involved in the computing aspects of the evaluation of fundamental mathematical functions.