Apply one step of the Gauss Seidel method to $A\textbf{x} = b$ with A = $\begin{bmatrix} 4 & 2 & 1 \\ 1 & 4 & 1 \\ 1 & 2 & 4 \end{bmatrix}$, b = $\begin{bmatrix} 4\\ 5\\ 8 \end{bmatrix}$ $x_{0}$ = $\begin{bmatrix} 64\\ 64\\ -128 \end{bmatrix}$ Does the method converge to the solution?
To answer the question, this is what I did...
$$x_{1} = \frac{4 - 2x_{2} - x_{3}}{4} => x_{1} = \frac{4-2(64)+128}{4} = 1$$ $$x_{2} = \frac{5-x_{1}-x_{3}}{4} = \frac{5-1+128}{4} = 33$$ $$x_{3} = \frac{8-x_{1}-2x_{2}}{4} = \frac{8-1-66}{4} = -59/4$$
Did I do it right? If not I am confused. Can someone help me do this correct? Thanks.
Yes, your first iteration is correct. Here are some iterates and details.
Iteration 1:
$$\left( \begin{array}{cccccccccc} 1. \\ 33. \\ -14.75 \end{array} \right)$$
Iteration 2: $$\left( \begin{array}{cccccccccc} -11.8125 \\ 7.89063 \\ 1.00781 \end{array} \right)$$
Iteration 3:
$$\left( \begin{array}{cccccccccc} -3.19727 \\ 1.79736 \\ 1.90063 \end{array} \right)$$
... Iteration 10: $$\left( \begin{array}{cccccccccc} 0.208341 \\ 0.812503 \\ 1.54166 \end{array} \right)$$
You can compare this against the actual result of:
$$x = \left( \begin{array}{c} \frac{5}{24} \\ \frac{13}{16} \\ \frac{37}{24} \\ \end{array} \right) = \left( \begin{array}{c} 0.208333 \\ 0.8125 \\ 1.54167 \\ \end{array} \right)$$