The Logarithmic Error and Newton's Method for the Square Root The problem of obtaining optimal starting values for the calculation of the square root using Newton's method is considered. It has been pointed out elsewhere that if relative error is used as the measure of goodness of fit, optimal results are not obtained when the initial approximation is a best fit. It is shown here that if, instead, the so-called logarithmic error is used, then a best initial fit is optimal for both types of error. Moreover, use of the logarithmic error appears to simplify the problem of determining the optimal initial approximation. CACM February, 1969 King, R. F. Phillips, D. L. square root, Newton's method, relative error, logarithmic error, best fit, optimal approximation, maximal error, recurrence relation, integer root, error curve 5.12 5.13 CA690206 JB February 20, 1978 10:55 AM 1440 4 1932 1932 4 1932 1932 4 1932 2094 4 1932 2159 4 1932 962 5 1932 1566 5 1932 1832 5 1932 1932 5 1932 1932 5 1932 1932 5 1932 1999 5 1932 2159 5 1932 962 6 1932 1932 6 1932 1932 6 1932 1932 6 1932