[SI-LIST] Re: spectral domain vs BEM (was non-negative......)
- From: Steve Corey <steve@xxxxxxxxxxxxxx>
- To: si-list@xxxxxxxxxxxxx
- Date: Mon, 10 Sep 2001 11:15:43 -0700
Chris -- I was hoping an EM person might respond to this, but I'm sure it's
true. From a general algorithms perspective, it's the same problem as the
non-zero off-diagonals. For example, in a diagonally dominant C matrix -- after
a certain number of orders of magnitude below the magnitude of the diagonal, the
result has lost precision due to round-off in the computer, which is incapable
of computing an exact solution to most matrices. Since any digits beyond that
point are meaningless, neither Cij nor Cji is exactly correct, so either one, or
the average of the two, is just as valid as the other. A typical approach would
be to use the one with the smaller magnitude, to help preserve the positive
definiteness of the matrix.
If you're designing an algorithm with the knowledge that computer solutions are
inherently inexact, you generally assume that if someone is depending on your
nth decimal place for their design to function, then it's probably not going to
function anyway once it gets fabbed. Whether this is healthy pragmatism or just
arrogance probably depends on who's designing the algorithm -- in most cases, I
like to think it's the former. 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:
> 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
> >
>
> ------------------------------------------------------------------
> To unsubscribe from si-list:
> si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field
> For help:
> si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field
>
> List archives are viewable at:
> //www.freelists.org/archives/si-list
> or at our remote archives:
> http://groups.yahoo.com/group/si-list/messages
> Old (prior to June 6, 2001) list archives are viewable at:
> http://www.qsl.net/wb6tpu
>
------------------------------------------------------------------
To unsubscribe from si-list:
si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field
or to administer your membership from a web page, go to:
//www.freelists.org/webpage/si-list
For help:
si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field
List archives are viewable at:
//www.freelists.org/archives/si-list
or at our remote archives:
http://groups.yahoo.com/group/si-list/messages
Old (prior to June 6, 2001) list archives are viewable at:
http://www.qsl.net/wb6tpu
Other related posts:
