From mboxrd@z Thu Jan 1 00:00:00 1970 From: Jonathan Leto To: Francesco Potorti` Cc: gsl-discuss@sources.redhat.com Subject: Re: Root finding page manual suggestion Date: Wed, 19 Dec 2001 13:20:00 -0000 Message-id: <20011018210751.A13280@leto.net> References: X-SW-Source: 2001/msg00621.html Just as a word of warning, I found a interesting site called "Why not use Numerical Recipes?", written by JPL. Link: http://math.jpl.nasa.gov/nr/ It seems that a few professional numerical analysts have found quite a few things wrong with much of the code and theory. Francesco Potorti` (pot@gnu.org) was saying: > 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. -- jonathan@leto.net "Wir mussen wissen. Wir werden wissen."