[SI-LIST] Re: spectral domain vs BEM (was non-negative......)
- From: Chris Cheng <chris.cheng@xxxxxxxxxxxx>
- To: si-list@xxxxxxxxxxxxx
- Date: Tue, 4 Sep 2001 18:08:09 -0700
while on this subject, i want to add a little more dirt.
a lot of commerical EM tools generate non-symmetric
off diagonal elements ( Cij ne Cji). they just take
the average and report it in Cij and Cji. this is
even more common than non negative C elements.
-----Original Message-----
From: Steve Corey [mailto:steve@xxxxxxxxxxxxxx]
Sent: Tuesday, September 04, 2001 4:26 PM
To: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Re: spectral domain vs BEM (was non-negative......)
This is correct -- you are comparing two stable matrix inversion algorithms.
LU decomposition with full pivoting is stable for any well-conditioned
matrix,
whereas ICCG (if I remember right, someone please correct me if I'm
propagating
falsehoods) is stable for symmetric positive definite matrices. Each one
croaks when passed a matrix which is nearly singular. LU tends to compute
huge
values, ICCG fails to converge. Poorly conditioned matrices are the result
of
finite word length in the computer (e.g., a double can only carry ~16 digits
of
decimal precision), and both algorithms must face that fact.
The primary reason for using an iterative matrix inversion algorithm is to
try
and speed up the inversion of certain classes of matrices, but if the method
is
correctly applied, it doesn't sacrifice accuracy. (On a practical note, to
speed up convergence, the criteria of an iterative method may often be
loosened
to have it quit before it converges as tightly as it possibly can. This
could
also explain the lack of precision in Ray's results.) A decent treatment of
LU
decomposition and also of the family of conjugate gradient algorithms is
given
in "Matrix Computations" by Golub and van Loan.
Independent of the matrix solution algorithm employed, certain formulations
of
a particular EM problem (FEM, BEM, spectral domain, etc.) will lead to more
stable matrices to invert than other formulations will, resulting in better
precision in the final answer. This is where I recuse myself, since I
haven't
coded up any of the above methods since graduate school. This is why every
design group needs an EM computations guru -- to ask which formulation to
apply
to which problem 8^).
-- Steve
-------------------------------------------
Steven D. Corey, Ph.D.
Time Domain Analysis Systems, Inc.
"The Interconnect Modeling Company."
http://www.tdasystems.com
email: steve@xxxxxxxxxxxxxx
phone: (503) 246-2272
fax: (503) 246-2282
-------------------------------------------
Chris Cheng wrote:
> the point i was trying to say was :
> i was always told that in FEM, you can't explicitly pivot the
> n diagonal sparse matrix and have to use ICCG while in
> BEM you can use LU since the matrix is small but dense. i
> thought the underlying assumption is explicit is exact while
> implicit is converge so explicit is a better choice. given we
> see round off error in LU, maybe we should rethink the argument.
> i particular the round off gets propagated forward with LU while
> in ICCG the round off is reset in every interation.
> chris
>
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