Question

Employ the Gauss-Seidel method, solve the system. 10𝑥 + 𝑦 + 𝑧 = 12 2𝑥 + 2𝑦 + 10𝑧 = 14 2𝑥 + 10𝑦 + 𝑧 = 13

Answer 1

Rewrite

$$10x+y+z=12$$

$$2x+10y+z=13$$

$$2x+2y+10z=14$$

$$x_{n+1}=\dfrac{1}{10}(12-y_n-z_n)$$

$$y_{n+1}=\dfrac{1}{10}(13-2x_{n+1}-z_n)$$

$$z_{n+1}=\dfrac{1}{10}(14-2x_{n+1}-2y_{n+1})$$

Initial gauss $(x,y,z)=(0.5,0.5,0.5)$

1^st Approximation

$$x_{1}=\dfrac{1}{10}(12-0.5-0.5)=1.1$$

$$y_{1}=\dfrac{1}{10}(13-2(1.1)-0.5)=1.03$$

$$z_{1}=\dfrac{1}{10}(14-2(1.1)-2(1.03))=0.974$$

2^nd Approximation

$$x_{2}=\dfrac{1}{10}(12-1.03-0.974)=0.9996$$

$$y_{2}=\dfrac{1}{10}(13-2(0.9996)-0.974)=1.00268$$

$$z_{2}=\dfrac{1}{10}(14-2(0.9996)-2(1.00268))=0.999544$$

3^rd Approximation

$$x_{3}=\dfrac{1}{10}(12-1.00268-0.999544)=0.9997776$$

$$y_{3}=\dfrac{1}{10}(13-2(0.9997776)-0.999544)=1.00009008$$

$$z_{3}=\dfrac{1}{10}(14-2(0.9997776)-2(1.00009008))=1.000026464$$

Solution

$x=1.000, y=1.000, z=1.000$