Re: NEC-LIST: A note on Sparse Iteration.

> I have done a little more work with the Sparse Iteration Method and thought
> the results might be of interest to those who write their own code. To
> review, the equation
>
> (1) ZI=V
> (2) (S+D)I=V, SI=V-DI
> (3) SI_(k+1)=V-DI_k

this only works well when Z is effectively diagonal dominant which
is only the case when you have
a) may short unconnected wires
b) arrays of wiregrids
c) helices as one segment comes very close to another one

and additionally all segments should have the same length!

it is not efficient for general 3D cases, complicated wire structures
like meshed cars, ships, planes etc.

For special cases, the proposed method can be quite fast. More
often it is not even converging (for the interested: the
largest eigenvalue of S^(-1)(-D) < 1, see Golub, van Loan "Matrix
Computation" page 511).

if you compare "exact" (I take this to be LU factorization) and the
iterative method you really have to be careful to compare
a good LU and the iterative method. If you write un inefficient
LU like the one that is in the usual NEC as it comes you
end up with a poor "exact" time, and the iterative is often
overestimated.

> Tabe 2. Wire and dielectric. The no. of unknowns is 1733.
>
> Exact solution time =10.56 min.

My test with LAPACK and Atlas on a PII Mobile, 233 MHz:
694 unknowns 5.7 s
2923 unknowns 425 s
This does not really comfortably compare with 10 min for 1733
unknowns (I project this rather to be 88 sec or 1 min 28 sec).

> Nr Ns T
> 70 100 4.46
> 100 25 1.22
> 200 25 1.603

25 is a high number of iterations. Take a Krylov subspace
method and precondition with D and you end up with
only a couple of iterations, typically 5 or 6.

> though there are strong off -diagonal terms. Since the physical structure is
> coiled, there is strong coupling between segments that are far apart in index.

these are better words than mine :)

> I expect that simply making S a banded matrix would work in the case of an
> unsymmetric Z, since the efficacy really depends on the diagonal dominance.
> It would be straightforward to modify an LU factorization for a banded case,
> and it probably already exists.

http://www.netlib.org/lapack

cheers!
juergen

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