Generalized logarithmic error and Newton’s method for the $m$th root.

Abstract:

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 for each positive integer $m$, $m \geqq 3$, and each negative integer $m$, there is a certain generalized logarithmic error for which a best initial fit to 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.