MATLAB MATLAB MATLAB

MATLAB

Note: This problem is identical to the Jacobi method problem, except that you need to use the Gauss-Seidel method. Consider the linear system A, D = p, where A, is an nxn matrix with 2's on the main diagonal, -1's directly above and below the main diagonal and O's everywhere else. For example, 1 2 -1 0 0 0 -1 2 -1 0 0 As = 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 This is a discretized version of Poisson's equation d$(x) = p(x), which appears very often in physical applications. We will discuss discretizations and differential equations, including the origin of the matrix A., later in the class. Construct the matrix A103 in Matlab. (You should be able to do this in only a few lines with the help of the diag command. In particular, you should figure out what the commands diag(v), diag(v, 1) and diag(v, -1) do when v is a vector.) Save your matrix in a variable named A. In addition, construct the 103 x 1 vector p such that the jth entry of p is defined according of the formula P;=2(1 - cos (172)) sin (161) Save this vector in a variable named rho. If you are particularly good with trigonometric identities, you can show that the exact solution to our problem is the 103 x 1 vector whose jth entry is defined according to the formula 0; = sin 7a Save this vector in a variable named phi. The Gauss-Seidel method for this problem can be written as = M0X-1 +c. (Note that means the kth guess for and it is an entire vector. It does not mean the kth entry of .) Find the largest in magnitude) eigenvalue of M and save the magnitude of this eigenvalue in a variable named ans1. (Remember, you can use the abs function to find the magnitude of a number. Your answer should be a positive, real number.) Use the Gauss-Seidel method to solve for o. Your initial guess should be a 103 x 1 vector of all ones, and you should use a tolerance of 10-5. That is, you should stop when - Pi-illo