proceedings

FORUM PROCEEDINGS
Mr. Ferber: If you want to solve a small set of equations
with seven or eight unknowns, the machines aren't
adapted to working with such a small number of cards
and one comes to the point where it is quicker to do them
some other way. Then again, since the work goes up very
rapidly with increasing order, you soon come up against a
blank wall in that the work is prohibitive. How do you
handle the problem?
Mr. Bell: We do enough matrix work to keep the basic
602 panels wired up ; so we need not allow time for that.
Our operators are familiar with the job, and they handle
it efficiently even with only seven or eight unknowns.
Everything is kept ready for them. To give you some idea
of time, it requires between one and two hours to evaluate
an eighth order complex determinant. I f there are enough
problems, you can split them up so that some are mUltiplying
while others are being sorted and gang punched, with
no idle machines.
We have tried to handle special cases by special procedures,
but have lost on it. I feel it is better not to use
special procedures. Instead we stick to methods that all of
our people understand; then we are more certain of coming
out with a good answer.
Mr; Harman: I was wondering if you could get around
the division step in your formula by the expansion of a
second-order minor, which involves the difference between
two multiplications.
Mr. Bell: We have tried doing that, and here is the.problem
we got into: the numbers change tremendously in
size, and it is necessary to stop for inspection. That takes
longer than the regular method.
Dr. Tukey: The thing is to get away from division by
small numbers.
Mr. Bell: Yes. 1\1:ost of our work is brought in to us.
Sometimes we know what is behind the problems, and
sometimes we do not. Most of the work is in engineering
fields where extreme accuracy is not required. But we do
have to calculate with expanded accuracy occasionally.
Dr. Tttkey,' As I understand the situation, the pessimists
thought-and I was one of them-that you lost, roughly
speaking, a constant number of significant figures for each
new equation. Now it has been proved! that it doesn't go
like that; the loss goes like the logarithm of the order.
If you are thinking of big equations that is a tremendous
improvement. I think that size ought not to be taken as
grounds for pessimism.
REFERENCE
1. J. VON NEUMANN and H. H. GoLDSTINE, "Numerical Inverting
of Matrices of High Order," Bull. Amer. Math. Soc., 53 (1947),
pp. 1021-99.
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