Program using the program MATHEMATICA the Jacobi Method which are described in full on page 34 of the text.

Make sure your initial vectors match those on page 34.

Check that you get the same output. i.e. your relative normed errors match those listed on page 34.

Use the Power Method that we described at the end of class to check that the eigenvalues for (I-BA) match the eigenvalues outputted by the Mathematica function Eigenvalues.

Finally, use x = LinearSolve[A, b] to get the "actual" solution. And then compute relative normed error as discussed in class.

Page 34

Suppose A is a 50*50 matrix. This is so small that there is no difficult computing the actual solution via Gauss-Jordan elimination, and then the actual error for the Jacobi process.

Set A to be tridiagonal matrix with entries _i,i = 0.5 on the diagonal _{i, i-1} = _i+1 = 0.25 on the super diagonal and sub diagonal and zeros elsewhere. Next, take b to have i^th entry equal to 1/i. The spectral radius if I-BA is approximately 0.998<1. We expect the Jacobi method to converge slowly. Indeed, this is the case. Starting with x₀ = (1, 1, , 1) we have relative normed error of 1.101438 after 50 iterations, 0.41144 after 500 iterations and 8 x 10^-5 at 5,000 iterations.