Example 1: Solving a system of equations by the Gauss-Seidel method
Use the Gauss-Seidel method to solve the system
4x1 + x2 - x3 = 3
2x1 + 7 x2 + x3 = 19
x1 - 3 x2 +12 x3 = 31
<=> x1 = -1/4 x2 + 1/4 x3 + 3/4
x2 = -2/7 x1 - 1/7 x3 + 19/7
x3 = -1/12 x1 + 1/4 x2 + 31/12

Solution:

We have

= 0 -1/4 1/4
-2/7 0 -1/7
-1/12 1/4 0
x1
x2
x3
+ 3/4 = G +
19/7
31/12
The iteration formulas are
<=> x1(k+1) = -1/4 x2(k) + 1/4 x3(k) + 3/4
x2(k+1) = -2/7 x1(k+1) - 1/7 x3(k) + 19/7
x3(k+1) = -1/12 x1(k+1) + 1/4 x2(k+1) + 31/12
The difference between the Gauss-Seidel method and the Jacobi method is that here we use the coordinates x1(k),...,xi-1(k) of x(k) already known to compute its ith coordinate xi(k).

If we start from x1(0) = x2(0) = x3(0) = 0 and apply the iteration formulas, we obtain
k x1(k) x2(k) x3(k)
0 0 0 0
1 0,75 2,50 3,15
2 0,91 2,00 3,01
3 1,00 2,00 3,00
4 1,00 2,00 3,00
The exact solution is: x1 = 1, x2 = 2, x3 = 3.

For instance, when k=2, we have x2(2)= 2,00:
x2(2) = -2/7 x1(2) - 1/7 x3(1) + 19/7 = -2/7 · 0,72 - 1/7 · 3,15 + 19/7 = 2,00


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