Generalized Logarithmic Error and Newton ' s Method for the rath Root

The problem of obtaining optimal starting values for the calculation of integer roots using Newton's method is considered. It has been shown elsewhere that if relative error is used as the measure of goodness of fit. then optimal results are not obtained when the initial approximation is a best fit. Furthermore, if the so-called logarithmic error instead of the relative error is used in the square root case, then a best initial fit is optimal for both errors It is shown here that lor each positive integer m, m g 3, and each negative integer m. there is a certain generalized logarithmic error for which a best initial fit io the mth root is optimal. It is then shown that an optimal fit can be found by just multiplying a best relative error fit by a certain constant. Also, explicit formulas are found for the optimal initial linear fit. Introduction. The logarithmic error S is defined as Ô = ln(l + Ô), where ô is the relative error. In [1] it is shown that if the logarithmic error is used instead of the relative error, then an initial fit to the square root that minimizes the maximum logarithmic error also minimizes the maximum logarithmic error after one or more Newton iterations. Furthermore, this best initial fit minimizes the maximum relative error after one or more Newton iterations. It is also noted that this nice property of the logarithmic error does not hold for mth roots, m = 3, 4, 5, ..., and m = 1, — 2,_The reason that negative values of m are included here is that for negative m the Newton iteration involves only multiplication and subtraction. For machines with very slow division time, for example ILLIAC IV, it might be faster to compute 1/x as x~ ' and x1"" as x(x~ ' T~ '. It is the purpose of this note to show that for a certain generalized logarithmic error a best initial fit to the mth root minimizes the maximum relative error as well as the generalized logarithmic error after one or more Newton iterations. It will also be shown that a best generalized logarithmic fit can be obtained by simply multiplying a best relative fit by a certain constant. Generalized Errors and Optimal Initial Fits. We will use the notation ô for the relative error and Ö for another error that can be written as ó = f(ô). Let y0 be an initial approximation to the mth root of x and y„ be the nth Newton iterate. Then if ôn is the relative error after n Newton iterations, y — x1"" 1 àn = " i,m—, (« = 0,1,2,...), v„+1=x m and (1) c5„+1 = -[(m1)(1 + <5„) + (l +¿„)'-m] 1 e^ Received October 21, 1968. revised May 15, 1969. A MS Subject Classifications. Primary 6520. 4140.