Civil Engineering Reference
In-Depth Information
In the
iteration‚ one could use the
equation to update the value
of
x
as follows:
.
Alternatively‚ observing that if these computations are carried out sequen-
tially‚ at the time when
is to be computed‚
is known
we may
use the following update formula instead:
.
The update formula in Equation (2.70) is referred to as the Gauss-Jacobi
method‚ while the formula in Equation (2.71) is called the Gauss-Seidel method.
The former has the advantage over the latter of being easier to parallelize‚ since
each variable in the iteration may be simultaneously updated; this
is not possible for the latter due to the sequential dependencies between the
update formulæ for and
While there are various ways of characterizing the requirements for conver-
gence‚ it can be shown that if the matrix
A
is diagonally dominant
10
‚ both the
Gauss-Seidel and Gauss-Jacobi methods converge unconditionally‚ regardless
of the initial guess. This is a particularly useful fact‚ since there are practical
circuit topologies for which the nodal or MNA matrix is diagonally dominant.
The successive overrelaxation (SOR) method is an alternative update for-
mula that sets the next update value for a variable to be
where is the updated value using the Gauss-Seidel update formula
from Equation 2.71. This formula uses an extrapolation that finds the update
as a weighted sum of the previous iterate and the Gauss-Seidel update. It
is clear that for this method is identical to the Gauss-Seidel method.
Moreover‚ it has been proven that if
(0,2), the procedure will not converge.
