Use the GaussSeidel method to solve the following linear sys

Use the Gauss-Seidel method to solve the following linear system.

3x₁ - x₂ + x₃ = 1,

3x₁ + 6x₂ + 2x₃ = 0,

3x₁ + 3x₂ + 7x₃ = 4.

Solution

Let\'s apply the Gauss-Seidel Method to the system from Example 1:

3x₁ - x₂ + x₃ = 1,

3x₁ + 6x₂ + 2x₃ = 0,

3x₁ + 3x₂ + 7x₃ = 4.

At each step, given the current values x₁^(k), x₂^(k), x₃^(k), we solve for x₁^(k+1), x₂^(k+1), x₃^(k+1) in

3x₁^k+1 - x₂^k + x₃^k = 1,

3x₁^k+1 + 6x₂^k+1 + 2x₃^k = 0,

3x₁^k+1 + 3x₂^k+1 + 7x₃^k+1 = 4.

To compare our results from the two methods, we again choose x⁽⁰⁾ = (0, 0, 0). We then find x⁽¹⁾ = (x₁⁽¹⁾, x₂⁽¹⁾, x₃⁽¹⁾)by solving

3x₁¹ - 0 +0 = 1,

3x₁¹ + 6x₂¹ + 0 = 0,

3x₁¹ + 3x₂¹ + 7x₃¹ = 4.

Let us be clear about how we solve this system. We first solve for x₁⁽¹⁾ in the first equation and find that

x₁⁽¹⁾ = 1/3

We then solve for x₂⁽¹⁾ in the second equation, using the new value of x₁⁽¹⁾ = 1/3, and find that

x₂⁽¹⁾ = [0 - 3(1/3)] / 6 = -1/6.

Finally, we solve for x₃⁽¹⁾ in the third equation, using the new values of x₁⁽¹⁾ = 1/3 and x₂⁽¹⁾ = -1/6, and find that

x₃⁽¹⁾ = [4 - 3(1/3) ? - 3(1/6)] / 7 = 7/2=3.5

The result of this first iteration of the Gauss-Seidel Method is

x⁽¹⁾ = (x₁⁽¹⁾, x₂⁽¹⁾, x₃⁽¹⁾) = (1/3, -1/6, 3.5).