Please explain the reasoning behind your answer

This is for the subject Numerical Methods

You are given the results of the following Matlab commands and some related spy plots. size(A) ans = 407 407 symchk = norm(A-A', Inf) symchk = 5.5511e-16 Anrm = norm(A, Inf) Anrm = 473.7444 AInrm = norm(inv(A), Inf) AInrm = 3.5799 p = symr cm(A) Spy plot of A Spy plot of A(p.p) 0 100 100 200 200 300 300 400 0 400 0 100 300 400 100 300 400 200 nz = 5826 200 nz = 5826 Cholesky factor of A Cholesky factor of A(pp) 0 0 100 100 200 200 300 300 400 0 400 0 400 100 400 100 200 300 nz = 15724 200 300 nz = 8505 (a) [2 marks] What are the steps to solve Ax = b using the Cholesky factorization A = RTR? Solve RTy = b for y by forward substitution, and then solve Rx = y for x by back substitution. Solve RTy = b for y by back substitution, and then solve Rx = y for x by forward substitution. O Solve Ry = b for y by back substitution, and then solve RTX = y for x by forward substitution. (b) [2 marks] Calculate the condition number of the matrix A in co-norm. 3.580 1696 O 473.7 (b) [2 marks] Calculate the condition number of the matrix A in o-norm. O 3.580 1696 0 473.7 (c) [2 marks] Calculate the sparsity of A as a percentage. O 3.52% O 96.48% 0.04% (d) (4 marks] The elements of the coefficient matrix A are known to 8 significant figures and the elements of the right-hand-side vector bare known exactly. i) Give an estimate on the relative error in the computed solution to Ar = b. 8E-6 O 2E-16 0 5E-9 ii) How many significant figures do you have in the computed solution? 4 OOO 2 6