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# Linear Algebra Anyone

David Doshay doshay at ursa.arc.nasa.gov
Sun May 17 18:30:22 EST 1992

```In article <24445 at life.ai.mit.edu>, lyle at substantia-nigra.ai.mit.edu (Lyle J. Borg-Graham) writes:
|>
|>
|> In article <1992May16.170522.25590 at news.arc.nasa.gov> doshay at ursa.arc.nasa.gov (David Doshay) writes:
|> >You mention large dense matrix problems.
|> >   ...
|> > The big difference is that my matrix is quite sparse. When
|> >the elements are in an unbranched cable, the system is tri-diagonal,
|> >but branches make for complications. We also have 2d tissue connecting
|> >to 1d cables, and that causes large departures from tri-diagonal.
|> >
|>
|> Hines (Efficient computation of branched nerve equations, Int. J.
|> Bio-Medical Computing, 1984, V.15, pp.69-76) describes an elegant
|> reordering of nodes for branched (tree) topologies that transform the
|> general sparse matrix into a tri-diagonal one. Several neuron
|> simulators that I am aware of use this technique (including one in
|> Lisp that I use). This technique is also discussed by Mascagni
|> (Numerical Methods for Neuronal Modeling, Chapter 13 in Methods in
|> Neuronal Modeling, MIT Press/Bradford Books, 1989, ed. Koch and
|> Segev).
|>

The Hines method is great when you have branched cables, as it is a
reordering method that treats each part of the martix as locally tri-
diagonal, and then efficiently solves the tridiagonal parts in a good
order to preserve this property overall, because the solution of tri-
daigonal matricies with the right method is far more efficient than
general methods on these special matricies.

In my case this does not work because the 2d tissue that we also model
is not tridiagonal at all, though it can be properly ordered to be strongly
diagonally dominant. We could use the Hines method on all the branched
cables, using other methods when we do the 2d tissue, but at this point
we are using a sparce matrix solver that is relatively fast without adding
the complication of using 2 different methods for different parts of the
total matrix. We will probably add a preconditioner, that does what is
essentially a generalization of the Hines technique, when our data sets
get larger. At this point we have not yet seen that it will speed us up
enough to implement it.

David 		doshay at ursa.arc.nasa.gov

the thought police insist I tell you:
my thoughts, not NASA's

```