Third Order Convergence for Square Roots Using Newton's Method

[ttw]

Wikipedia gives a simple quartic method from the Bakhtshali Manuscript. It's nice in that it gives rational approximations and could be used on polynomials. It's actually two steps of Newton's method with algebraic simplifications. (Newton's method is nice for square roots because the higher-order derivatives disappear; on the other hand, this makes memoryless high-order methods tricky.)

To get Sqrt(S), one uses two auxiliary variables, A and B.

A=(S-X^2)/2S

B=X+A

X(new)=B-(A^2)/2B equivalent to (X+A)-(A^2)/2*(X+A)

I suppose one could algebraically compound two of these.

The interval of convergence [/quote]isn't given.