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Whereas our principal mathematical justification for an iterative
averaging comes from the mean value property, the behavior of the
iteration in terms of convergence must ultimately come from the properties
of the numerical methods. In this section we will look closer at the
mathematical details of our iterative processes with the ultimate goal of
finding the optimal SOR coefficient.
We can generalize the Jacobi method by differentiating for boundary conditions,
Here
selects those values which are non-boundary values, and
indicates
those which are boundary conditions.
The above equation can be generalized by writing, for the Jacobi case,
![$\displaystyle Dx^{n+1}=Bx^n + b$](img29.png) |
(3.4) |
For the GS case, we use the new values as they become available whereby,
The reader can convince themselves that L is a lower triangular
matrix and U an upper triangular, where the Jacobi form could be simplified
as, for example,
![$\displaystyle \left(\begin{array}{cccc}D&U&U&U L&D&U&U L&L&D&U L&L&L&D\end{array}\right)\left(\begin{array}{c}X_1 X_2 X_3 X_4\end{array}\right)$](img31.png) |
(3.5) |
That L is a triangular matrix and D is uniformly nonzero along the
diagonal means that the determinant of D-L will not be zero, whereby
D-L is invertible. Thus we have,
SOR has us calculate the difference between the new value of the
potential we are calculating as recommended by GS, and the old value
pre-calculation. Calling this difference Q, SOR then adds wQ to the
old coefficients, where w is the SOR coefficient to be determined for
optimal progress (
is simply GS,
is SOR). Thus in matrix
form, the SOR method can be written
And obvious substitution is
from which we get
![$\displaystyle x[k+1]=Hx[k]+(D-wL)^{-1}wb$](img42.png) |
(3.8) |
The reader is encouraged to view reference [2],[3]
for a more detailed exposition, as we take more than a few liberties of
omitting detail here.
Subsections
Next: Conditions for the the
Up: Simultaneous (successive) Over Relaxation
Previous: Jacobi, Gauss-Seidel Method
Timothy Jones
2006-02-24