Jacobi Method

Description

The Jacobi method is an iterative method to solve systems of linear equations. The iterative methods have the advantage over elimination method of less round-off errors. It differs from the Gauss-Seidel method because it uses zeroes for the initial approximations.

Algorithm

1. If the diagonal elements are non-zero, each equation is rewritten for the corresponding unknown:

   $$\begin{aligned} x_1 = \frac{b_1 − a_{12}x_2 − a_{13}x_3 - \cdots − a_{1n}x_n}{a_{11}} \\ x_2 = \frac{b_2 − a_{21}x_1 − a_{23}x_3 - \cdots - a_{2n}x_n}{a_{22}} \\ \vdots \\ x_n = \frac{b_n − a_{n1}x_1 − a_{n2}x_2 − \cdots − a_{nn−1}x_{n−1}}{a_{nn}} \end{aligned}$$

2. Make all the variables equal to zero, $x_1 = 0$, $x_2 = 0$, …, $x_n = 0$.

3. Use the rewritten equations to calculate the new estimates.

4. Calculate the relative error for each $x_i$, if it’s small enough, the iterations stop:

   $$|ep|_i= ∣\frac{x^{new}_i − x^{old}_i}{x^{new}_i}∣ \times 100$$