Heads up for a hot new root seeking algorithm!!

[Namir]

(01-19-2017 03:23 AM)Claudio L. Wrote:  

(01-18-2017 11:53 PM)Namir Wrote:  I am not sure that Ostrowski's method has an order 4 of convergence. Any reference that confirm this?

Namir

The paper I cited, shows at the very top on page 78 (it's actually the third page) that plain Newton has order 2, Ostrowski has order 4, and another formula has order 8. It says "it's well established", so I guess it must be true.

Yes the paper states, on page 78, that Ostrowski's convergence is of order 4. The author is mistaken!! I tested 24 functions and compared Newton, Halley, Ostrowski, and my new Ostrowski-Halley method. The methods of Halley and Ostrowski often found the root in the same number of iterations. If Ostrowski's method is order 4, I didn't see it in the test results. My new algorithm did, in general a little bit better than Halley and Ostrowski (after all it's a combination of these two methods).

Many mathematicians have, somewhat recently, improved on Ostrowski's method to yield algorithms with high to very high convergence rates. That's nice and dandy, but it comes at the cost of very high number of function calls, making the simpler (and slower converging) algorithms more practical if one regards the number of function calls as the cost of doing business in finding roots.

Namir