LM Assignment 2 3NMNLA4 Numerical Linear Algebra Instructions: Please upload your handwritten solutions by 21 April 1pm. Please include both your name and your student ID on the front page. Plagiarism check: Your submission should be your own work and should not be identical or substantially similar to other submissions. A check for plagiarism will be performed on all submissions. Assessment: This sheet is assessed, with a maximum contribution to your final mark of 5%. 1. Let A _ R n_ have full rank and let W _ R n_ be symmetric and positive definite. Let F(y) = 1 2 kb _ Ayk 2 W_1 ,

LM Assignment 2 3NMNLA4 Numerical Linear Algebra Instructions: Please upload your handwritten solutions by 21 April 1pm. Please include both your name and your student ID on the front page. Plagiarism check: Your submission should be your own work and should not be identical or substantially similar to other submissions. A check for plagiarism will be performed on all submissions. Assessment: This sheet is assessed, with a maximum contribution to your final mark of 5%. 1. Let A _ R n_ have full rank and let W _ R n_ be symmetric and positive definite. Let F(y) = 1 2 kb _ Ayk 2 W_1 , where b _ R n is given. (a) Find the Hessian of F. Hence, show that F is strictly convex. Deduce that F has a unique minimiser x in R m which satisfies the augmented system of equations W A AT O q x = b 0 , q = W_1 (b _ Ax). [8] In the following, the augmented system will also be denoted by Ku = f. (b) The iterative method below is used to construct a sequence of approximations to u. ( Wq k+1 = b _ Ax k , x k+1 = x k + _AT q k+1 , for some parameter _ 0 to be chosen suitably. Show that the above iteration corresponds to a splitting of the form K = M _ N, i.e., it is an iteration of the form Mu k+1 = Nu k + f, u k = q k x k , for some matrices M, N which you should identify. [4] (c) Find the iteration matrix G = M_1N and show that it has n zero eigenvalues, with the remaining m given by the eigenvalues of S = I _ _ATW_1A. [5] (d) Let B = ATW_1A. Show that _(S) = 1 _ __(B), where the notation _(S), _(B) denotes eigevalues of S and B, respectively. Deduce that the spectral radius _(G(_)) is minimised for _ _ = 2 _1(B) + _m(B) . [4] (e) Let A, W, Tm be defined as follows: A = 1 _ m 1m _ Tm, W = Im _ Tm, Tm = tridiag[1, 6, 1]. Show that B = Tm and deduce that lim m__ _ _ = 1 6 . Hence, find the limit of _(G(_ _ )) as m _ _. What can you say about the number of steps required to satisfy a given tolerance as m _ _? [For this part, you may want to use the expression given in Q12, Examples 5, for the eigenvalues of a Toeplitz tridiagonal matrix T = tridiag[a, b, c].] [9

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