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Successive Over Relaxation - SOR Method

Successive Over Relaxation - SOR Method

Background

Suppose that iteration is used to solve the linear system , and that is an approximate solution. We call the residual vector, and if is a good approximation then . A method based on reducing the norm of the residual will produce a sequence that converges faster. The successive over relaxation - SOR method introduces a parameter which speeds up convergence. The SOR method can be used in the numerical solution of certain partial differential equations.

Consider that the n�n square matrix A is split into three parts, the main diagonal D, below diagonal L and above diagonal U. We have A = D - L - U.

[Graphics:Images/SORmethodMod_gr_8.gif] =

--

Definition (Diagonally Dominant). The matrix is strictly diagonally dominant if

for .

Theorem ( Jacobi Iteration ).  The solution to the linear system    can be obtained starting with  , and using iteration scheme


where

          and  .

If   is carefully chosen a sequence is generated which converges to the solution  P,  i.e.  .
A sufficient condition for the method to be applicable is that A is strictly diagonally dominant or diagonally dominant and irreducible.

Theorem ( Gauss-Seidel Iteration ).  The solution to the linear system    can be obtained starting with  , and using iteration scheme


where

          and  .

If   is carefully chosen a sequence is generated which converges to the solution  P,  i.e.  .
A sufficient condition for the method to be applicable is that A is strictly diagonally dominant or diagonally dominant and irreducible.

Theorem ( SOR Iteration ).  Given a value of the parameter    (chosen in the interval ),  the solution to the linear system    can be obtained starting with  , and using iteration scheme


where

         [Graphics:Images/SORmethodMod_gr_40.gif]   and  .

If   is carefully chosen a sequence is generated which converges to the solution  P,  i.e.  .
Remark.  A theorem of Kahan states that the SOR method will converge only if    is chosen in the interval   .

Remark. When we choose the SOR method reduces to the Gauss-Seidel method.

Mathematica Subroutine (Jacobi Iteration).

Mathematica Subroutine (Gauss-Seidel Iteration).

Mathematica Subroutine (Successive Over Relaxation).

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