[SI-LIST] Re: spectral domain vs BEM (was non-negative......)
- From: Steve Corey <steve@xxxxxxxxxxxxxx>
- To: si-list@xxxxxxxxxxxxx
- Date: Tue, 04 Sep 2001 16:26:28 -0700
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
>
> -----Original Message-----
> From: Ray Anderson [mailto:Raymond.Anderson@xxxxxxx]
> Sent: Tuesday, September 04, 2001 2:50 PM
> To: si-list@xxxxxxxxxxxxx
> Subject: [SI-LIST] Re: spectral domain vs BEM (was non-negative......)
>
> Mick-
>
> I was responding to Chris's query about which algorithm the
> tool used. He was wondering about which inversion method might
> have been used.
>
> I agree, whether the off-diag elements are non-positive or not
> depends on the definition of the matrix, but we jumped topics
> to discuss another aspect.
>
> -Ray
>
> >
> >
> >
> >>
> >>Looks like you are discussing the mathematical aspects of different
> >>algorithms. But I think the
> >>original question is about the physical aspects. Should the off-diagonal
> >>elements be non-positive
> >>or non-negotive ? Fortunately, we can answer it directly based on the
> >>definitions, independent of any numerical examples. It does not matter
> how
> >>you calculate them. So, we can judge the mathematical results. It is very
> >>true that all algorithms have drawbacks which lead to numerical
> >>difficulties and errors.
> >>
> >>Mick
> >>
> >>
> >>-----Original Message-----
> >>From: Ray Anderson [mailto:raymonda@xxxxxxxxxxxxxxxxxxxxxxx]
> >>Sent: Tuesday, September 04, 2001 4:39 PM
> >>To: si-list@xxxxxxxxxxxxx
> >>Subject: [SI-LIST] spectral domain vs BEM (was non-negative......)
> >>
> >>
> >>
> >>One of the Apsim guys may want to comment, but I believe that the
> >>Apsim RLGC tool that I was using implements the spectral domain
> >>method which I think is similar to BEM in some respects but
> >>different in others. (I'm going to stop right here, I only
> >>know enough about this to be dangerous :)
> >>
> >>RLGC can optionally use the FEM, but it is recommended that FEM be
> >>utilized for more complex structures that are defined with multiple
> >>dielectrics than simple stripline constructs.
> >>
> >>
> >>-Ray
> >>
> >>
> >>
> >>>
> >>>
> >>>which brings into another question,
> >>>ray,
> >>>was the tool using FEM (most likely using ICCG) or
> >>>BEM (most likely using LU)?
> >>>chris
> >>>
> >>
> >
>
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