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 ((7) |

sqroot
Convergence plots for different
recursive algorithms, shown in Table 1.
Figure 2 |

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 .

Convergence | ||

Muir | 1 | linear |

Secant | x_{n-1} |
quasi-quadratic |

Newton | x_{n} |
quadratic |

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.

4/20/1999