Heads up for a hot new root seeking algorithm!!

[JurgenRo]

(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.

You have to distinguish between local- and global error. Plain Newton is simply Tayler series Expansion of 1st order:

f(x_(n+1)) = f(x_n) + hf'(x_n) + (1/2)h^2f''(ceta),

where h= x_(n+1)-x_n, ceta in (x_n+1,x_n) and R:=(1/2)h^2f''(ceta) is the term defining the local error (of each step). That is, locally plain Newton is o(h^2), i.e. quadratic. The global error though (the accumulated error of all N steps ) is o(Nh^2)=o(h), i.e. linear behavior. Normally you are interested in the total accumulated error, i.e. in the global error. So to my understanding plain Newton is a linear scheme.

Juergen