Assignment 2 Fall 2013 DUE at the beginning of your Lab 1. Consider the following system of linear equations 2×1 + x2 ?? x3 ?? x4 = 11 x2 ?? x3 + 2×4 = 7 x1 ?? x2 + 6×3 + x4 = ??10 x1 ?? 2×3 + x4 = 4 By hand, solve this linear system using Gaussian elimination with partial pivoting and backward substitution. Use fractions throughout your calculations. 2. Let A = 2645 3 ??1 1 5 ??2 ??1 2 1 375 (a) Compute by hand the LU factorization of A, without pivoting. Verify your result in MATLAB. (b) Applying the LU factorization from (a), compute the solution of Ax = b where b = [1;??21;??9]T . (c) Applying the LU factorization from (a), compute the rst column of A??1. (d) With the LU factorization from (a), compute the determinant of A. 3. Consider A = 26664 1 2 3 4 ??1 1 2 ??1 ??2 5 13 3 1 ??1 ??10 2 37775 (a) Find kAk1 and kAk1 (by hand). (b) Compute A??1 and the Frobenius norms kAkf and kA??1kf in MATLAB. (c) Use MATLAB to nd kAk2, and kA??1k2. Applying these results, calculate the condition number in norm-2 of A. Verify your result in MATLAB using the cond command. Do you expect a system Ax = b to be ill-conditioned? 4. Consider the linear system of equations 7×1 + 2×3 = 2 x1 ?? 5×2 ?? x3 = 1 x2 + 4×3 = 1 (a) Write down the components of the Jacobi iteration equation. Do not apply any itera- tions, just write down the iteration mechanism. (b) Modify the MATLAB code provided to solve the system by the Jacobi iteration. Use an initial guess of [0;??0:5; 0]T , and iterate until the approximate percent relative error is less than 0:05%. Provide a printout of your MATLAB code and a printout of the output created. Give the number of iterations required and the approximate solution to the system. (c) Starting with an initial vector x(0) = [0;??0:5; 0]T apply, by hand, the Gauss-Seidel iteration until the approximate percent relative error is less than 5%.
Recommended List (not to be handed in) Chapter Sections Questions 8 8.2 8.2, 8.4, 8.9 9 9.2-9.4 (omit 9.3.2) 9.6, 9.7 ,…