141_Gauss-Seidel Method

The Gauss-Seidel method is a numerical technique for solving systems of linear equations. It involves splitting the corresponding matrix of the system into two triangular matrices and then iteratively computing the solution until it converges.

 

Given a system with $n$ equations, it is described as follows.

 

 

Assuming a set of $3 \times 3$ equations where all diagonal elements are not zero, each $x$ can be solved as follows.

 

 

You can start the solution process by selecting initial estimates for $x$. The initial valuses can simply be set to zero. First, substitute 0 into the first equation to calculate a new value for ${x_1}$, then substitute this value into the second equation, assuming ${x_3}$ is still zero. Similarly, calculate the third equation. Then, return to the first equation and repeat the entire process until the values sufficiently converge to the true values. Solve this through an example.

 

 

The True solutions is 

 

 

First, solve each equation for the unknown on the diagonal.

 

 

Assuming ${x_2}$ and ${x_3}$ are zero, the first equation can be calculated as follows.

 

 

Substitute the calculated value and ${x_3}$ into the second equation to compute it.

 

 

Substitute these values into the third equation to complete the first iteration.

 

 

And the second iteration is as follows.

 

 

Therefore, through this method, it can be observed that the values are converging towards the actual solution. Additional iterations can be performed to further refine the accuracy of the values.