Solving system of equations by using the Gauss-Seidel method

Question

I'm having problem solving this system of equations:

\begin{array}{lcl}x & -3y & 100z & = & 123 \\ 200x & -3y & 2z & = & 765 \\ x & -500y & 2z & = & 987\end{array}

I'm supposed to find the solution to the system of equations with accuracy of $\epsilon=0.01$ using the Gauss-Seidel method. $(3.5;\;-2;\;1)$ is the starting point. Also, the convergence needs to be guaranteed. I will appreciate any help. I hope everything I wrote is understandable.

Answer 1

The Gauss-Seidel Method requires the matrix to be in diagonally dominant form.

This matrix is not diagonally dominant and G-S does not converge (sometimes it still may).

The first step is to put the matrix in D-D form so we have $Ax = b$ as

$$A = \begin{pmatrix}200 & -3 & 2 \\ 1 & -500 & 2 \\ 1 & -3 & 100 \\ \end{pmatrix}, b = \begin{pmatrix}765\\ 987 \\ 123 \\ \end{pmatrix}$$

You can figure out how many iterations you need to meet your accuracy requirement

$$\begin{align}x_0 &= (3.5,-2,1)\\ x_1 &= (3.785,-1.96243,1.1332771)\\ x_2 &= (3.784230779,-1.96189843,1.133300739)\\ x_3 &= (3.784238516,-1.96189832,1.133300665)\\ x_4 &= (3.784238519,-1.96189832,1.133300665)\end{align}$$

The exact results are

$$x = \begin{pmatrix} \dfrac{900785}{238036} \\ -\dfrac{3269017}{1666252} \\ \dfrac{3776729}{3332504} \\ \end{pmatrix} \approx \begin{pmatrix} 3.784238518543414\\-1.961898320302091\\1.133300665205503 \end{pmatrix}$$