If we consider the general form of the square root iteration
we can estimate the convergence rate by the difference between the actual estimation at step (n+1) and the analytical value . For the general case, we obtain
Figure 2 Convergence plots for different recursive algorithms, shown in Table 1.
The possible selections for from Table 1 clearly show that the recursions described in the preceding subsection generally have a linear convergence rate (that is, the error at step n+1 is proportional to the error at step n), but can converge quadratically for an appropriate selection of the parameter , as shown in Table 7. Furthermore, the convergence is faster when is closer to .
We therefore conclude that Newton's iteration has the potential to achieve the fastest convergence rate. Ideally, however, we could use a fixed which is a good approximation to the square root. The convergence would then be slightly faster than for the Newton-Raphson method, as shown in Figure 2.