|Re: New root-seeking algorithm|
Message #6 Posted by Namir on 10 Dec 2005, 9:24 a.m.,
in response to message #5 by Patrick
The Bisection method is a reliable method despite it's slowness. Often, complex root-seeking implementation use a combination of Newton (or similar/better methods) with he Bisection as a backup plan in case Newtons method yield guesses that stray away.
The Quartile method is a root-bracketing method that is better than the Bisection. The value for alpha can be in the range of 0.25 and 0.3. Th Quartile algorithm uses a slightly different approach--compares the absolute values of the functions at interval [A. B]. The Quartile methd is not a variant of Bisection. Te Bisection is a special case of the Quartile method (setting alpha to 0.5).
The method of False-Position is the next higherup root-bracketing method that uses the function values as weights. This algorithm does not systematically shrink the interval containing the root. To stop iterating, you compare the current new guess with the last new guess.
If you are going to resort to the Bisection method as the main or backup method, then you can replace it with the Quartile method to have fewer iterations. The reduction in iterations depends on the function and the initial root-bracketing interval.
Newtons method is popular in root solving. It uses calls to teh functio and it's first derivative. The Haily method is generally better than Newton's. It calls the function and its first two derivatives. The Householder method is even faster, but calls n he fucntion and it's first three derivatives! By using function calls to approximate the derivatives, I found that Haily's method is in general the optimum method (considering iterations and function calls).
Parick, I welcome your suggstions for root-seeking algorithms (both variants of other methods or original algorithms). I think the subject is fascinating and I am interested in all sugegstions.
Edited: 10 Dec 2005, 9:28 a.m.