Root finding page manual suggestion

[Francesco Potorti`]

In the "Root  Finding Algorithms using Derivatives" page  one reads that
the Newton's method converges  quadratically for single roots, while the
secant  method  has 1.6  convergence  order,  and  "can be  useful  when
computation of the derivative is costly".

In fact, as far as I know, it is almost always preferable to Newton's
method.

Quoting from "Numerical Methods" by Germund Dahlquist and Ake Bjorck,
translated by Ned Anderson - Prentice-Hall Inc., 1974.

Chapter 6.4.1. (the end) pg 229
  The choice  between the secant  method  and  Newton-Raphson's method
depends on the amount of work required to  compute f'(x).  Suppose the
amount  of work to  compute f'(x)  is T  times  the amount of  work to
compute a value of f(x).   Then an  asymptotic analysis can be used to
motivate the rule: if T > 0.44, then use the secant method; otherwise,
use Newton-Raphson's method.

I've used this  criterion in some small numerical  program I've written,
and it  works ok.   The above  means that Newton's  method wins  only if
computing f'(x) is  more than twice faster than  computing f(x), a quite
rare occurrence in practice.  I suggest this fact to be mentioned in the
manual.

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