# Large sets of Linear Equations

Whenever a person eagerly inquires if my computer can solve a
set of 300 equations in 300 unknowns, I must quickly suppress the
temptation to retort, “Yes, buy why bother?” There
*are*, indeed, legitimate sets of equations that large. They
arise from replacing a partial differential equation on a set of
grid points, and the person who knows enough to tackle this type of
problem also usually knows what kind of computer he needs. The odds
are all too high that our inquiring friend is suffering from a
quite different problem: he probably has collected a set of
experimental data and is now attempting to fit a 300-parameter
model to it—by Least Squares! The sooner this guy can be
eased out of your office, the sooner you will be able to get back
to useful work—but these chaps are persistent. They have
fitted three-parameter models on desk machines with no serious
difficulties and now the electronic computer permits them more
grandiose visions. They leap from the known to the unknown with a
terrifying innocence and the perennial self-confidence that every
parameter is totally justified. It does no good to point out that
several parameters are nearly certain to be competing to
“explain” the same variations in the data and hence the
equation system will be nearly indeterminate. It does no good to
point out that *all* large least-squares matrices are
striving mightily to be proper subsets of the Hilbert
matrix—which is virtually indeterminate and
uninvertible—and so even if all 300 parameters were
beautifully independent, the fitting equations would still be
violently unstable. All of this, I repeat, does no good—and
you end up by getting angry and throwing the guy out of your
office.

Most of this instability is unnecessary, for there is usually a
reasonable procedure. Unfortunately, it is undramatic, laborious,
and requires thought—which most of these charlatans avoid
like the plague. They should merely fit a five-parameter model,
then a six-parameter one. If all goes well and there is a
statistically valid reduction of the residual variability, then a
somewhat more elaborate model may be tried. Somewhere along the
line—and it will be much closer to 15 parameters than to
300—the significant improvement will cease and the fitting
operation is over. There is no system of 300 equations, no 300
parameters, and no glamor. But a person has to know some
statistics, he has to have a clear idea about the mechanisms by
which the variability has entered his data, and he has to know the
intended use for his fitted formula. It is infinitely easier to let
a computer try to solve 300 equations and hope to put some sort of
interpretation on the numbers, assuming one gets any, and be safe
from criticism because the computer did it all. It is difficult to
choose mathematical models to represent material phenomena and,
while it is not easy to evaluate the parameters, the real
difficulties are *statistical*, not computational. The
computer center’s director must prevent the looting of
valuable computer time by these would-be fitters of many
parameters. The task is not a pleasant one, but the legitimate
computer users have rights, too. The alternative commits everybody
to a miserable two weeks of sloshing around in great quantities of
“Results” that are manifestly impossible, with no
visible way of finding the trouble. The trouble, of course, arises
form looking for a good answer to a poorly posed problem, but a
computer director seldom knows enough about the subject matter to
win any of those arguments with the problem’s proposer, and
the impasse finally has to be broken by violence—which
therefore might as well be used in the very beginning.