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From: john@spectre.unm.edu (John Prentice)
Newsgroups: comp.lang.fortran
Subject: Re: Need complex matrix solver for CM
Message-ID: <1991Jun27.233111.958@ariel.unm.edu>
Date: 27 Jun 91 23:31:11 GMT
References: <1991Jun27.161918.10218@leland.Stanford.EDU> <1991Jun27.190724.26161@ariel.unm.edu> <7887@mace.cc.purdue.edu>
Organization: Dept. of Math & Stat, University of New Mexico, Albuquerque
Lines: 51

In article <7887@mace.cc.purdue.edu> bfp@mace.cc.purdue.edu (Bryan Putnam) writes:
>In article <1991Jun27.190724.26161@ariel.unm.edu> john@spectre.unm.edu (John Prentice) writes:
>>In article <1991Jun27.161918.10218@leland.Stanford.EDU> francis@hanauma.stanford.edu (Francis Muir) writes:
>>>Re-write the code in REAL; it will probably illuminate.
>>
>>This is an amazingly presumptuous, not to mention useless, comment.  If I
>
>That sounds like a perfectly good solution to me, although it makes the
>linear system twice as large. If you need a High School math reminder:
>
>Ax=b ==>
>
>+ A(r)  -A(i) +  x(r)          b(r)
>|             |           =    
>+ A(i)   A(r) +  x(i)          b(i)


Thanks, but I know perfectly well how to convert a complex linear system
to a real linear system.  As I said before, there are reasons I don't wish
to do so having to do with the structure of the matrix I am working with.
If you really need an example of where one might not wish to convert
a complex to a real linear system, consider a diagonally dominant
banded complex matrix.  Look at your real matrix above.  Both the 
A(r) and A(i) block matrices will now be diagonally dominant and banded,
but the big matrix will not be.  By converting this to a real matrix,
you lose both the diagonal domiance and you introduce off diagonal bands.  
Both make solving the system alot harder with anything but Gaussian 
elimination.  My systems are order 10,000 to 100,000.  You can't solve
them with straight Gaussian elimination and expect to get the answers back
anytime soon.  That means you need iterative schemes and these often
depend on the structure of the matrix such as the bandedness and diagonal
dominance.  One could construct schemes that would work with the
real version of the matrix, but why do it if software exists to do the problem
as is?  It just makes the problem obscure.  In my case, I am solving my 
system using an iterative scheme which at some point exploits the fact 
that my matrices have diagonal bands which are large compared to the 
rest of the matrix.  I solve the smaller banded problem using Gaussian 
elimination as part of my iterative scheme.  Reworking the algorithms and 
proving their convergence properties is not worth the effort if complex 
matrix solvers exist, as they in fact appear to do.

What bugged me about the original comment was not the suggestion to convert
to a real matrix, it was the comment "it will probably illuminate".  In
either case, my question has been answered, so perhaps we can let this
particular discussion end.

John
-- 
John K. Prentice    john@spectre.unm.edu (Internet)
Computational Physics Group
Amparo Corporation, Albuquerque, NM