Consider the linear system A x = b, where

(a) First, solve the equation directly by Gaussian Elimination.
(b) Using the initial approximation x(0) = 0, carry out three iterations of the Jacobi algorithm to compute x(l), x(2) and x(3). How close are you to the exact solution?
(c) Write the Jacobi iteration in the form x(k+l) = T(k) + c. Find the 3 × 3 matrix T and the vector c explicitly.
(d) Using the initial approximation x(0) = 0, carry out three iterations of the Gauss-Seidel algorithm. Which is a better approximation to the solution--Jacobi or Gauss-Seidel?

(f) Determine the spectral radius of the Jacobi matrix T, and use this to prove that the Jacobi method iteration will converge to the solution of A x = b for any choice of the initial approximation x(0).

(h) For the faster method, how many iterations would you expect to need to obtain 5 decimal place accuracy?
(i) Test your prediction by computing the solution to the desired accuracy.

4 2