Page 42 - Matrix Analysis & Applied Linear Algebra

34 Chapter 1 Linear Equations

If two straight lines are almost parallel and if one of the lines is tilted only
slightly, then the point of intersection (i.e., the solution of the associated 2 × 2
linear system) is drastically altered.
L'
L

Perturbed
Solution
Original
Solution

Figure 1.6.1

This is illustrated in Figure 1.6.1 in which line L is slightly perturbed to
become line L . Notice how this small perturbation results in a large change

in the point of intersection. This was exactly the situation for the system given
in Example 1.6.1. In general, ill-conditioned systems are those that represent
almost parallel lines, almost parallel planes, and generalizations of these notions.
Because roundoff errors can be viewed as perturbations to the original coeffi-
cients of the system, employing even a generally good numerical technique—short
of exact arithmetic—on an ill-conditioned system carries the risk of producing
nonsensical results.
In dealing with an ill-conditioned system, the engineer or scientist is often
confronted with a much more basic (and sometimes more disturbing) problem
than that of simply trying to solve the system. Even if a minor miracle could
be performed so that the exact solution could be extracted, the scientist or
engineer might still have a nonsensical solution that could lead to totally incorrect
conclusions. The problem stems from the fact that the coefficients are often
empirically obtained and are therefore known only within certain tolerances. For
an ill-conditioned system, a small uncertainty in any of the coefficients can mean
an extremely large uncertainty may exist in the solution. This large uncertainty
can render even the exact solution totally useless.
Example 1.6.2

Suppose that for the system

.835x + .667y = b 1
.333x + .266y = b 2
the numbers b 1 and b 2 are the results of an experiment and must be read from
the dial of a test instrument. Suppose that the dial can be read to within a