Given any a > 0, we consider the following nonlinear equation: 13 - a' =0. Our goal is to find a fast fixed-point iteration that converges to the root r* = a. a) Consider the following iteration: "+1 = g(xx), g(x) :=+ +x-0. (2) Show that * = a is the only fixed-point of this fixed-point iteration. For an arbitrary initial point ro a, will this iteration converge to r* = a? Explain why! Hint: Write g(x) - a in the form g(x) -a = h(x)(x - a) for some suitable function h. What can you say about |h(x)| or |h(a;)|? b) Consider the following fixed point iteration: Th+1 = g(xx), g(x) := (3)- Show that * = a is the only fixed point of the mapping g. Show that g(x) > a for all r > 0. - Show that the iteration (3) converges to r* = a for any initial point ro e (0, co). What is the rate of convergence