ITERATIVE SOLUTION METHODS:
           aims, tools, craftmanship


The cooling of enfant's brains, the pollution
of groundwater, the forces over a spacerocket,
the flows of the ocean, and electronic devices,
represent only a few of the examples that have in
common that after modeling very large
linear systems have to be solved. Very large
nowadays may mean in the order of one billion of
unknowns or more. This may seem an impossible task,
but many of these problems have been
solved successfully by iterative solution methods.
Modern methods include popular algorithms such as
Conjugate Gradients, GMRES, and Bi-CGSTAB.
These are examples of
the large family of Krylov subspace methods.

However, besides many  reports of great success
we see also many reports of failure, so that the
natural question is how and when they can be used
effectively. The answer to this question
requires some basic insight in how they work
and in the circumstances where they may be expected to
fail. This depends strongly on spectral pro-
perties of the matrix of the system and the
way to improve them is preconditioning.
We will discuss these aspects and we will see
how we can monitor the effects of precondi-
tioning, and the necessity for further tuning,
with the help of the readily available
iteration parameters.


                         Henk van der Vorst
                         Utrecht University