The rate of convergence of Newton's process

Summary

The author applies the method of nondiscrete mathematical induction (see [2–5]) which involves considering the rate of convergence as a function, not as a number, to Newton's process and proves that the rate of convergence is

$$\omega (r) = \frac{{r^2 }}{{2(r^2 + d)^{1/2} }}$$

whered is a positive number depending on the initial data (see Theorem 2.3).