Some Remarks on the Computation of $\sqrt X$

Abstract

For a as an approximation to $\sqrt X$, the Newton formula $: = \tfrac{1}{2}(a + A/a)$ defines the (improved) value aa as the mean value of two approximations a and A/a. This is first examined using simple geometry. Next some approximations are obtained by replacing $\tfrac{1}{2}$ by a factor p. For example aa := p(a + A/a) and $aa: = p \cdot a + \tfrac{1}{2} \cdot (A/a)$ with p = p(a,A) slightly less than $\tfrac{1}{2}$. From this we are led to some ‘formula generating formulas’, where iteration is used to obtain new iteration formulas. Next some division-free formulas for $\sqrt X$ are examined. Finally it is shown how the same iteration technique as for square roots can be used to obtain approximations for functions like sin, using an operator (integral) equation instead of the algebraic equation used for $\sqrt X$.