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The iteration converges quadratically starting from any real initial guess
*a*_{0} except zero. When *a*_{0} 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

| |
(6) |

so, for example if the error is one significant digit at one
iteration, at the next iteration it is two digits, then four, etc. We
cannot use equation (6) in place of the Newton
iteration itself, because it uses the answer to get the
answer *a*_{t+1}, and also we need the factor of proportionality.
Notice, however, if we take this factor to be 1/(2*a*_{t}),
then cancels and equation (6) becomes
itself the Newton iteration (3).
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

| |
(7) |

therefore
| |
(8) |

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).

** Next:** Minimum phase factors
** Up:** Theory
** Previous:** Spectra factorization
Stanford Exploration Project

7/5/1998