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A third iterative method, called the Successive Overrelaxation (SOR) Method, is a generalization of and improvement on the GaussSeidel Method. Here is the idea:
For any iterative method, in finding
Here is how we derive the SOR Method from the GaussSeidel Method. First, notice that we can write the GaussSeidel equation as
so that
We can subtract
Now think of this as the GaussSeidel correction
where, as we just found,
and where generally
Written out in detail, the SOR Method is
We can multiply both sides by matrix D and divide both sides by ω to rewrite this as
then collect the
When we solve for x^{(k+1)}, we get
Notice that the SOR Method is also of the form
so optimal convergence is achieved by choosing a value of ω that minimizes
As we did earlier for the Jacobi and GaussSeidel Methods, we can find the eigenvalues and eigenvectors for the
In practice, we typically use a computer to perform the iterations, so we need implementable algorithms in order to use these methods for
Method  Algorithm for performing iteration k + 1: for i = 1 to n do: 

Jacobi  
Gauss Seidel 

SOR 
David M. Strong, "Iterative Methods for Solving [i]Ax[/i] = [i]b[/i]  The SOR Method," Loci (July 2005)
Journal of Online Mathematics and its Applications