QUESTION 4 (a) (1) When does the power method for finding the largest eigenvalue of a non-singular matrix A not converge, or converges very slowly? (ii) How can you speed up the power method to find the largest eigenvalue A in fewer iterations of the approximate value of us known a (8) ds (b) Consyder the characteristic value problem dy + 2ky = 0, y(0) = (3) = 0 Il wu apply the method of hmte differences to this problem, with A2 =1 we get the linear equation 2k-2 1 2k-2 where y, and y2 denote the approximate values of y at a = 1 and 1 = 2 The matrix 2)()-(3) 1-2 has the eigenvalues 2 = -1 and 12 = -3 (i) According to this what are the two approximate values of k foz which the boundary valuc problem (*) has non-zero solutions? (11) What are the approximate non-zero solutions corresponding to cach of the characteristic values in (b)) above 1 QUESTION 4 (a) (1) When does the power method for finding the largest eigenvalue of a non-singular matrix A not converge, or converges very slowly? (ii) How can you speed up the power method to find the largest eigenvalue A in fewer iterations of the approximate value of us known a (8) ds (b) Consyder the characteristic value problem dy + 2ky = 0, y(0) = (3) = 0 Il wu apply the method of hmte differences to this problem, with A2 =1 we get the linear equation 2k-2 1 2k-2 where y, and y2 denote the approximate values of y at a = 1 and 1 = 2 The matrix 2)()-(3) 1-2 has the eigenvalues 2 = -1 and 12 = -3 (i) According to this what are the two approximate values of k foz which the boundary valuc problem (*) has non-zero solutions? (11) What are the approximate non-zero solutions corresponding to cach of the characteristic values in (b)) above 1