Page 45 - Matrix Analysis & Applied Linear Algebra

1.6 Ill-Conditioned Systems 37

changes. If a radical change in the solution is observed for a small perturbation
to some set of coefficients, then you have uncovered an ill-conditioned situation.
If a given perturbation does not produce a large change in the solution, then
nothing can be concluded—perhaps you perturbed the wrong set of coefficients.
By performing several such experiments using different sets of coefficients, a
feel (but not a guarantee) for the extent of ill-conditioning can be obtained. This
is expensive and not very satisfying. But before more can be said, more sophisti-
cated tools need to be developed—the topics of sensitivity and conditioning are
revisited on p. 127 and in Example 5.12.1 on p. 414.
Exercises for section 1.6

1.6.1. Consider the ill-conditioned system of Example 1.6.1:

.835x + .667y = .168,
.333x + .266y = .067.
(a) Describe the outcome when you attempt to solve the system
using 5-digit arithmetic with no scaling.
(b) Again using 5-digit arithmetic, first row scale the system before
attempting to solve it. Describe to what extent this helps.
(c) Now use 6-digit arithmetic with no scaling. Compare the results
with the exact solution.
(d) Using 6-digit arithmetic, compute the residuals for your solution
of part (c), and interpret the results.
(e) For the same solution obtained in part (c), again compute the
residuals, but use 7-digit arithmetic this time, and interpret the
results.
(f) Formulate a concluding statement that summarizes the points
made in parts (a)–(e).

1.6.2. Perturb the ill-conditioned system given in Exercise 1.6.1 above so as to
form the following system:

.835x + .667y = .1669995,
.333x + .266y = .066601.

(a) Determine the exact solution, and compare it with the exact
solution of the system in Exercise 1.6.1.
(b) On the basis of the results of part (a), formulate a statement
concerning the necessity for the solution of an ill-conditioned
system to undergo a radical change for every perturbation of
the original system.