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From: csrdh@marlin.jcu.edu.au (Rowan Hughes)
Newsgroups: comp.arch
Subject: Re: massive linpack
Message-ID: <1991Jun8.055711.13457@marlin.jcu.edu.au>
Date: 8 Jun 91 05:57:11 GMT
References: <9106070135.AA02947@ucbvax.Berkeley.EDU>
Organization: James Cook University
Lines: 37

In <9106070135.AA02947@ucbvax.Berkeley.EDU> jbs@WATSON.IBM.COM writes:


>         Richard Lethin asks:
>At what size N for massive linpack does the accumulated numerical fuzz
>swamp the precision available in a double precision number?
>         Very large.  Typical relative error (in the vector norm sense)
>in the solution is n*c*e, where n is matrix size, c is condition number
>and e is machine epsilon (2**-52 for IEEE double).  Therefore if you
>want a solution accurate to 1 part in 1000 (10 bits) and your matrix
>has condition number less than 1000 (10 bits) your n can be 2**30 or

I would have thought  (N**2)*C*E was more appropriate.

>         Richard Lethin asks:
>Aren't really large problems of interest sparse matrices anyway?  Who
>holds the record for massive sparse matrix solves?

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
have no sensitivity to round-off errors since the original matrix is
used continuously in the solution. Successive Over Relaxation was
the most commonly used method until a few years ago (SOR). It had some
very severe restrictions on the types of matrix problems it could solve;
the diagonal dominance condition. Biconjugate gradient methods, using
Chebychev orthogonalilization, are now the rage. They're iterative,
but round-off errors can eventually destroy the solution. The largest
problem I've seen solved is for N=100,000. Bico does'nt have such
a severe restriction regarding diagonal dominance. Bico is well
suited to vector architectures; thats what it was designed for. It
can also work quite well on parallel machines.

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