A (possibly) new way of arriving at the Newton-Raphson method??

Question

I'd really like some opinions on this. I feel my reasoning in the later steps is quite weak. The results tell me I guess that alpha = beta at the stat point, and that the approximation becomes exact in this limit (obvious). What intrigues me, though, are the forms taken by rho at the stat point (rho star) ie the similarity with NR

URL to paper: http://www.academia.edu/26637694/On_a_Simple_Geometric_Interpretation_of_the_Newton-Raphson_Method

Thanks

Answer 1

You're starting with an interval and evaluating the function and its derivatives at the end points. This should lead you to the Interval Arithmetic and function enclosures. A Newton method for finding (enclosing) all (!) the roots in a given interval is called Interval Newton method. See Warwick Tucker's Validated Numerics: A Short Introduction to Rigorous Computations, 2011.