7. (10 points) fixed point iteration of implicit methods] Consider a predictor-corrector method (see Section 5.9.4) consisting of forward Euler as the predictor and backward Euler as the corrector, and suppose we make N correction iterations, i.e., we set , = U, +kf(U.) for j = 0, 1, N-1 U;+1 = Un +kf(0) end UH+1=UN Note that this can be interpreted as a fixed point iteration for solving the nonlinear equation Un+1 = Ur+kf(UM+1) of the backward Euler method. Since the backward Euler method is implicit and has a stability region that includes the entire left half plane, as shown in Figure 7.1(b), one might hope that this predictor-corrector method also has a large stability region. (a) Find the polynomial RN (2) such that Un+1 = Ry()U for arbitrary N. (b) Plot the stability region Sy of this method for N = 2, 5, 10, 20 (perhaps using plots.n from the author's website) and comment on any change in the size of the stability region. (c) Note that the fixed point iteration above can only be expected to converge for the test problem if |KA|