The iteration converges quadratically starting from any real initial guess a0 except zero. When a0 is negative, Newton's iteration converges to the negative square root. Quadratic convergence means that the square of the error at one iteration is proportional to the error at the next iteration
Even though we cannot estimate the rate of convergence by because we don't know the answer s, we can get an estimate of it by looking at the difference between the intermediate solutions at two consecutive steps. From (3), we can write
Another interesting feature of the Newton iteration is that all iterations (except possibly the initial guess) overestimate the ultimate square root. This is obvious from equation (6).