Research and Advances

A note on best one-sided approximations

In this note we consider the relationship between best approximations and best one-sided approximations for three different measures of goodness of fit. For these measures simple relationships exist between best approximations and best one-sided approximations. In particular it is shown that a best approximation and best one-sided approximation differ only by a multiplicative constant when the measure is the uniform norm of the relative error. In this case problems involving best one-sided approximations can be reduced to problems involving best approximations. The result is especially significant if one wants to numerically determine a best one-sided approximation, since algorithms exist for numerically determining best approximations when the measure is the uniform norm of the relative error (see, for example, [1]).

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Research and Advances

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 inital 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.

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