Question

2. (a) Suppose we have to find the root xof x); that is, we have to solve )0. Fixed-point methods do this by re-writing the equation in the form x·= g(x*) , and then using the iteration scheme : g(x) Show this converges (x-→x. as n→o) provided that K < 1 , for all x in some interval x"-a < x < x*+a ( a > 0 ) about the rootx 6 points] (b) Newton's method has the form of a fixed-point iteration, that solves f(x*) = 0 . Use a Taylor-series argument to derive the iteration x) N6 points] (c) write out the Newton method iteration to find the root x of the function f(x)=(x-x*),, for constant P21 show that kel-1-(l-P),-xi show that I'm-x1-11--k-xǐ. Show that Newton's method gives the exact answer in a single iteration if p 1. Show that the convergence is linear for p>1, and that the convergence becomes slower for bigger p values 8 points]

Answer 1

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