Page 43 - Matrix Analysis & Applied Linear Algebra

1.6 Ill-Conditioned Systems 35

tolerance of ±.001, and assume that values for b 1 and b 2 are read as . 168 and
. 067, respectively. This produces the ill-conditioned system of Example 1.6.1,
and it was seen in that example that the exact solution of the system is
(x, y)=(1, −1). (1.6.1)

However, due to the small uncertainty in reading the dial, we have that
.167 ≤ b 1 ≤ .169 and .066 ≤ b 2 ≤ .068. (1.6.2)

For example, this means that the solution associated with the reading (b 1 ,b 2 )=
(.168,.067) is just as valid as the solution associated with the reading (b 1 ,b 2 )=
(.167,.068), or the reading (b 1 ,b 2 )=(.169,.066), or any other reading falling
in the range (1.6.2). For the reading (b 1 ,b 2 )=(.167,.068), the exact solution is
(x, y) = (934, −1169), (1.6.3)

while for the other reading (b 1 ,b 2 )=(.169,.066), the exact solution is
(x, y)=(−932, 1167). (1.6.4)

Would you be willing to be the first to fly in the plane or drive across the bridge
whose design incorporated a solution to this problem? I wouldn’t! There is just
too much uncertainty. Since no one of the solutions (1.6.1), (1.6.3), or (1.6.4)
can be preferred over any of the others, it is conceivable that totally different
designs might be implemented depending on how the technician reads the last
significant digit on the dial. Due to the ill-conditioned nature of an associated
linear system, the successful design of the plane or bridge may depend on blind
luck rather than on scientific principles.

Rather than trying to extract accurate solutions from ill-conditioned sys-
tems, engineers and scientists are usually better off investing their time and re-
sources in trying to redesign the associated experiments or their data collection
methods so as to avoid producing ill-conditioned systems.
There is one other discomforting aspect of ill-conditioned systems. It con-
cerns what students refer to as “checking the answer” by substituting a computed
solution back into the left-hand side of the original system of equations to see
how close it comes to satisfying the system—that is, producing the right-hand
side. More formally, if
x c =( ξ 1 ξ 2 ··· ξ n )
is a computed solution for a system

a 11 x 1 + a 12 x 2 + ··· + a 1n x n = b 1 ,
a 21 x 1 + a 22 x 2 + ··· + a 2n x n = b 2 ,
.
.
.
a n1 x 1 + a n2 x 2 + ··· + a nn x n = b n ,