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Somewhat faster convergent than Jacobi.

Choose
(i. e. lower triangle):

Solving the set of *implicit* equations

is almost as simple as solving the *explicit* Jacobi equations.
Since the matrices
and
are triangular the only
change is that in the downward recursion 2.1 some of the
right hand terms refer to the present iteration instead of the
previous, ; however, each term is available as soon as it is required.

__EXAMPLE:__
With the same data as in the previous example we find the
first two improved solutions

The *convergence rate* of the GSR scheme is governed by the matrix

It can be shown that the spectral radius of
is given by

so that the rate of convergence is now

In our example
and
.

** Next:** 2.3.4 Successive Over-Relaxation (SOR)
**Up:** 2.3 Iterative Methods
** Previous:** 2.3.2 Jacobi Relaxation
* Franz J. Vesely Oct 2005*

See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001