Jacobi, Seidel, and SOR Iterative Methods for Solving Linear Equations

This article discusses three iterative methods for solving linear systems of equations: Jacobi, Seidel, and Successive Over-Relaxation (SOR). All three methods are used to solve linear systems of the form Ax=b, where A is the coefficient matrix and b is the constant vector.

1. Jacobi Iteration:

The Jacobi method is the simplest iterative technique. It treats each unknown variable as independent and updates its value iteratively using a formula until a desired accuracy is achieved.

The Jacobi iteration formula is:

$x_i^{k+1} = \frac{1}{a_{ii}} (b_i - \sum\limits_{j=1,j\neq i}^n a_{ij}x_j^k)$

Here, $x_i^k$ represents the value of the ith unknown variable in the kth iteration, $a_{ij}$ is the element in the ith row and jth column of the coefficient matrix A, and $b_i$ is the ith element of the constant vector b.

2. Seidel Iteration:

The Seidel method also employs an iterative formula, but it utilizes the most recently updated values of the unknowns during each iteration, enhancing convergence speed.

The Seidel iteration formula is:

$x_i^{k+1} = \frac{1}{a_{ii}} (b_i - \sum\limits_{j=1}^{i-1} a_{ij}x_j^{k+1} - \sum\limits_{j=i+1}^n a_{ij}x_j^k)$

In this formula, $x_i^{k+1}$ represents the value of the ith unknown variable in the (k+1)th iteration, while $x_j^{k+1}$ denotes the newly updated values of the jth unknowns in the same iteration.

3. Successive Over-Relaxation (SOR) Iteration:

The SOR method combines elements of both the Jacobi and Seidel methods, introducing a relaxation factor ω to further accelerate convergence.

The SOR iteration formula is:

$x_i^{k+1} = (1 - ω)x_i^k + \frac{ω}{a_{ii}} (b_i - \sum\limits_{j=1}^{i-1} a_{ij}x_j^{k+1} - \sum\limits_{j=i+1}^n a_{ij}x_j^k)$

In this formula, $x_i^{k+1}$ represents the value of the ith unknown variable in the (k+1)th iteration, and ω, the relaxation factor, falls within the range of (0, 2).

In summary, all three iterative methods – Jacobi, Seidel, and SOR – can be employed to solve linear systems of equations. Their respective convergence speeds and accuracy requirements vary, making it essential to choose the most suitable method based on the specific problem characteristics.