5. It is known that the following cubic equation m33m2+m3=0 has only one real root A1 which is also the largest one in modulus. (a) Explain how the power method can be employed to obtain an approximation of A1 [Hint: You may consider the 3 x 3 companion matrix] (b) Apply the power method with initial guess 1 T Xo=l111l for FOUR times to obtain an approximation of A1. Please provide the steps such as the approximated eigenvectors x1, x2, x3, X4 obtained in each iteration. (c) Apply Newton's method with initial guess 1:0 = 4 FOUR times to obtain an approxi- mation of the root A1 of the above cubic equation. Please provide the steps such as the approximations 1:1, $2,103, 34 obtained in each iteration. (20 Points) 5. (a) It can be shown that the characteristic polynomial of the matrix 0 3 A = 0 1 3 is det(1/ - A) = 13 -3x2 + x -3. So power method can be applied to obtain the largest eigenvalue (root) in modulus. (4 Points) (b) Thus we can apply the power method to matrix A as follows. Xo = [0.5774, 0.5775, 0.5774]T X1 = [1.7321, 0.0000, 2.3094]T r1 = [0.6000, 0.0000, 0.8000]T X2 = [2.4000, -0.2000, 2.4000] r2 = [0.7059, -0.0588, 0.7059]7 X3 = [2.1176, 0.0000, 2.0588]T r3 = [0.7170, 0.0000, 0.6971]] X4 = [2.0912, 0.0199, 2.0912]7 Then Al ~ | |Xall2 = 2.9575. (8 Points) (c) The Newton's scheme reads In - 31, + In - 3 Inti = In - 3x2 - 61 + 1 Co = 4.0000 $1 = 3.3200 T2 = 3.0481 $3 = 3.0013 CA = 3.0000 (8 Points)