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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