EXERCISE 4 (15pt] (a) Implement the Polar-Ribiere conjugate gradient method. Hint: Modify the descentLineSearch. m template from tutorial 2. Copy the relevant lines in your report. [2pt] (b) Apply Polar-Ribiere conjugate gradient method to minimise the function / : R' - R Try two initial points ag = (-1,2) and ro = (-1, -0.25)and set the tolerance tol = le-4. Plot the iterates over the function contours. State your choice of any relevant parameters. [2pt] (c) What is the main limitation of Polar-Ribiere method, do you observe it or any other problems in optimisation in (b), can you explain? [2pt] (d) Can global convergence be guaranteed for this problem or not and why? If not, can you modify your Polar-Ribiere solver to guarantee global convergence and how? Hint: What are the conditions under which global convergence can be proven for Polar-Ribiere? Implement the modification, rerun your solver and compare its behaviour to the one in (b). Paraphrase the relevant theoretical results. [5pt] (e) Investigate convergence of the Polar-Ribiere method in (b) a posteriori and include one rel- evant error plot. What are the empirical convergence rates and how did you obtain them? Do they agree with the theoretical predictions? Paraphrase the relevant theoretical results. [4pt]