From mboxrd@z Thu Jan 1 00:00:00 1970 From: Francesco Potorti` To: gsl-discuss@sources.redhat.com Subject: Root finding page manual suggestion Date: Wed, 19 Dec 2001 13:20:00 -0000 Message-id: X-SW-Source: 2001/msg00615.html 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.