What is successive over relaxation?

Successive Over-Relaxation (SOR) is a numerical method used to solve systems of linear equations that arise in various fields, including physics, engineering, and economics. The method is an extension of the Gauss-Seidel iteration method, where an over-relaxation factor is introduced to accelerate convergence.

In SOR, the system of equations is represented as a matrix equation, in which the unknown variables are the elements of a vector. The over-relaxation factor (w) is introduced to increase the rate of convergence, and it typically lies between 1 and 2. When w = 1, the method is identical to the Gauss-Seidel method. The iterative process used in SOR involves substituting the updated values of the vector into the original equations repeatedly until a solution is found.

SOR can be more efficient than other iterative methods since it updates each variable before moving on to the next, which can lead to faster convergence. However, it is important to choose an appropriate value for w to optimize the rate of convergence. If the value of w is too large, the method may not converge at all, while if it is too small, the method may be slower than other iterative methods.

The SOR method has applications in many fields, such as fluid dynamics, electromagnetics, and optimization. It is a popular technique due to its ease of implementation and effectiveness in solving linear equations.