Iterative Methods for Numerical Analysis

Jacobi Iteration Method

For a system

$$ax_1 + bx_2 + cx_3 = A$$

Rearrange for each \( x \):

$$x_1 = \frac{1}{a}(A -bx_2 -cx_3)$$
$$x_2 = \frac{1}{b}(A -ax_1 -cx_3)$$
$$x_3 = \frac{1}{c}(A -ax_1 -bx_2)$$

Start with initial guess for \(x_1, x_2, x_3 \). eg \( x_1 = x_2 = x_3 = 0\) and evaluate the above three equations for the first iteration, \( x_{1,1}, x_{2,1}, x_{3,1}\), which is then evaluated the same way to obtain \( x_{1,2}, x_{2,2}, x_{3,2}\). This can be repeated for a finite number of iterations or until the difference between each iteration is negligible.

Gauss-Seidel Method

In the previous example with the Jacobi method, values entered in the three equations are updated every iteration. Gauss-Seidel updates these values for every calculation. i.e. After calculating \( x_{1,1} \), the value used to calculate \( x_{2,1} \) is \( x_{1,1} \) instead of \( x_1 \). This also means that Gauss-Seidel cannot be calculated in parallel the way Jacobi can.

Convergence Conditions

The system of equations must be diagonally dominant. The elements on the main diagonal must be equal or larger than the sum of all other elements on that row.