🚀 go-pugleaf

In <4112@ssc-bee.ssc-vax.UUCP> carroll@ssc-vax (Jeff Carroll) writes:
>>  csrdh@marlin.jcu.edu.au (Rowan Hughes) writes:
>>Yes, large problems (N > 10,000) are typically sparse, and with strong
>>diagonal banding. These can only be solved with iterative methods, and
>>only on vector, or large parallel machines. Iterative methods usually
>There is at least one fallacy here. I have worked on at least one class
>of problem involving boundary integral methods which is both large by
>this definition and *dense*. We typically solve these by LU decomposition.
>I submit that if we can solve large dense problems by direct methods, we
>can certainly solve sparse problems of the same size by the same methods.

This is now only slightly related to architecture, but I'll give my 2c
worth.
If Jeff has to solve large (N>5000) dense problems, then he has a real
problem. In my area (fluid dynamics) the bulk of matrices, originating from
the Navier-Stokes eqns, are large (n=100,000 is common) and very sparse,
typically  25 non-zero elements per row. Unfortunately they often have
zero's on the diagonal (psi-zeta form of NS). These types of problems are
best solved by methods specifically designed for sparse banded matrices.
Direct methods (LU, Householder etc) will work quite well, but they
will never have the capabilities of routines like BICO. I'd be so bold
as to suggest that Jeff's type of problem is more the exception, than
the rule. The largest problems capable of actually being solved will
always be sparse.

Philisophical note:
BICO is one of the most elegant solution methods I've ever seen; its
far more powerful than SOR, and fits perfectly into vector and parallel
architectures with very little code change.

-- 
Rowan Hughes                                James Cook University
Marine Modelling Unit                       Townsville, Australia.
Dept. Civil and Systems Engineering         csrdh@marlin.jcu.edu.au