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Re: a funny capacitor.



At 10:28 PM 3/4/01 -0500, Ludwik Kowalski wrote:

The following matrix shows the Cij values for my funny capacitor.

I +2.86, -0.460, -1.15, -1.25 I
I -0.460, +2.86, -1.15, -1.25 I
I -1.15, -1.15, +3.26, -0.965 I
I -1.25, -1.25, -0.965, +3.46 I
....
No singularity was encountered

The matrix as written is pretty darn nearly singular!
-- Whereas the matrix elements of C as given above are
all of order 1, the matrix elements of its inverse
are all on the order of 100. This is a tell-tale sign
of an almost-singular matrix.

-- The only reason it is not exactly singular is because it
is not exact. It is the result of a numerical (finite element)
simulation. For that matter, I note that all the matrix
elements are given to 2 decimal places, except for the (3,4)==(4,3)
pair. If these are rounded from -.965 to -.96 to make them
consistent with the others, then the matrix becomes singular.

-- Also note that this roundoff / correction is required to make
the matrix comply with the charge conservation laws.

Bottom line: the singular matrix is the right answer.

I am emphasizing the success of
the inversion because the determinant of the Cij matrix which
John posted here three days ago was zero and the inversion
was no possible.

I'm not sure "success" is the right word, when a more-careful calculation
would have been less successful.

Inverting a singular matrix is like finding the intersection of two
parallel lines. If there is any inaccuracy in the description of the
lines, they will intersect, but it will be someplace far away, and it won't
mean anything.

For a better way to think about this situation, see
http://mailgate.nau.edu/cgi-bin/wa?A2=ind0103&L=phys-l&P=R1463

BTW: The procedure described there -- regularization -- comes in handy in
a wide range of situations; it's not restricted to funny capacitors.

Once again I would like to thank John for
the help he provided.

Glad to help. It's an interesting puzzle.

Note that in each row (of equation 2) the term on the diagonal
is larger than other terms.

It pretty much has to be larger (in magnitude) (or equally large, if there
is only one other nonzero element in the row).

This follows immediately from charge conservation plus the following
lemma: The diagonal elements are all positive, and the others are all
negative or zero.

Proving that lemma is the interesting part. You can do it using fancy
differential-equation theory. Qualitatively it's related to the fact that
a (static) drumhead or rubber sheet always lies "between" its supports; it
never overshoots.