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Root finding page manual suggestion
- To: gsl-discuss at sources dot redhat dot com
- Subject: Root finding page manual suggestion
- From: Francesco Potorti` <pot at gnu dot org>
- Date: Thu, 18 Oct 2001 15:53:39 +0200
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.
Please Cc to me while replying, as I am not subscribed to the list.