a) Use Newtons method (do this using Python) to calculate n, n R>0. You can choose an example of your choice if you wish. One choice would be n = 18. You can start with an initial guess of 4. Hint: Write down the problem in terms of a function f(x), x R in such a way that one of the roots of this function would correspond to the square root of n, i.e. x = n.

b) If ei denotes the error at step i of an iterative method and we have the property lim i e(i+1) ei = S < 1, (1) the method is linearly convergent, with rate S. Now consider the fixed point iteration method x(i+1) = g(xi). Using the above definition and assuming that g is continuously differentiable, g(r) = r and S = |g'(r)| < 1, prove that the fixed point iteration converges linearly w