Can a matrix that is not diagonally dominant be solved by Gauss-Siedel iteration?

Question

I'm working on the following problem:

Here's what I've done so far:
$$-2i_1+4i_2+i_3+0i_4=0$$ $$-i_1+0i_2+i_3+6i_4=-14$$ $$5i_1-2i_2-i_3+i_4=6$$ $$i_1+2i_2+6i_3-i_4=6$$ It can be seen that, when put in matrix form, the entire matrix isn't diagonally dominant. So my question here is, can I still solve this matrix or do I have to reduce down to the echelon form? Help!

Answer 1

Hint

Rearrange the equations so they are in diagonally dominant form before applying the Gauss-Seidel (G-S) Algorithm as shown in these notes.

After doing this, the system to solve using G-S is

$$\left( \begin{array}{cccc} 5 & -2 & -1 & 1 \\ -2 & 4 & 1 & 0 \\ 1 & 2 & 6 & -1 \\ -1 & 0 & 1 & 6 \\ \end{array} \right)\left( \begin{array}{c} i_1 \\ i_2 \\ i_3 \\ i_4 \\ \end{array} \right)\text{ = }\left( \begin{array}{c} 6 \\ 0 \\ 6 \\ -14 \\ \end{array} \right)$$

Performing the algorithm with a starting point of $I_0 = (0,0,0,0)^T$, after eleven iterations, it converges to the correct result of

$$i_1 = 2, i_2 = 1, i_3 = 0, i_4 = -2$$

Of course, they only want you to perform five iterations.