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2.3.1 Iterative Improvement
Let
be the exact solution of
,
and let
be an inaccurate (or estimated)
solution vector, such that
Inserting this into the given equation we find
which may be solved for
. (Use double precision!)
EXAMPLE:
From
we find, using the decomposition
the correction vector
Now interpret the improvement equation as an iterative formula:
Replace
on the left hand side by an easily invertible
matrix
close to
:
or
This procedure converges to the solution of
if
.
This is the case if all eigenvalues of the matrix
are situated within the unit circle.
Next: 2.3.2 Jacobi Relaxation
Up: 2.3 Iterative Methods
Previous: 2.3 Iterative Methods
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001