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From: jbs@WATSON.IBM.COM
Newsgroups: comp.arch
Subject: massive linpack
Message-ID: <9106070135.AA02947@ucbvax.Berkeley.EDU>
Date: 7 Jun 91 01:39:40 GMT
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         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
10**9.  This would require 10**18 storage and 10**27 ops to solve so
is well beyond current machines.  Note once you have any bits in your
answer you can obtain more by iterative improvement an order N*2 pro-
cess.
         Richard Lethin asks:
Aren't really large problems of interest sparse matrices anyway?  Who
holds the record for massive sparse matrix solves?
         People should try to avoid solving problems in ways which in-
volve large dense matrix solves simply because they are so expensive.
I suspect in many cases where large dense matrices are solved, a diff-
erent method would be faster but I have no real evidence for this.
         It is unreasonable to ask for record sparse solves since the
difficulty is strongly dependent on the structure of the problem.
                       James B. Shearer