(a) Do any of the non-diagonally dominant systems in Exercise 10.5.3 lead to convergent Jacobi schemes? Check the spectral radius of the Jacobi matrix.
(b) For the convergent systems in Exercise 10.5.3, starting with the initial guess x = y = z = 0, compute the solution to 2 decimal places by using the Jacobi method, and check your answer by solving the system directly by Gaussian Elimination.
In Exercise 10.5.3
Which of the following systems have a diagonally dominant coefficient matrix?
(a) 5x - y = 1
-x + 3y = - 1
(b) 1/2 x + 1/3 y = 1
1/5 x + 1/4 y = 6
(c) -5x + y = 3
-3x + 2y = -2
(d) -2x + y + z = 1
-x + 2y - z = -2
x - y + 3 z = 1
(e) - x + 1/2 y + 1/3 z = 1
1/3 x + 2y + 3/4 z = -3
2/3 x 1/4 y - 3/2 z = 2
(f) x - 2y + z = 1
-x + 2y + z = -1
x + 3y - 2z = 3
(g) - 4x + 2y + z = 2
-x + 3y + z = -1
x + 4y - 6z = 3