Consider the linear system
4x - y - z = 1
-x + 4y - w = 2
-x + 4z - w = 0
- y - z + 4w = 1.
(a) Find the solution by using Gaussian Elimination and Back Substitution.
(b) Using 0 as your initial guess, how many iterations are required to approximate the solution to within five decimal places using
(i) Jacobi iteration?
(ii) Gauss-Seidel iteration?
Can you estimate the spectral radii of the relevant matrices in each case?
(c) Try to find the solution by using the SOR method with the parameter co taking various values between .5 and 1.5. Which value of co gives the fastest convergence? What is the spectral radius of the SOR matrix?