Answer to Question #235891 in Linear Algebra for lavanya

Question #235891

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


1
Expert's answer
2021-09-13T17:26:18-0400

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)"

1st 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"

2nd 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"

3rd 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"

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