Page 46 - Matrix Analysis & Applied Linear Algebra

38 Chapter 1 Linear Equations

1.6.3. Consider the two straight lines determined by the graphs of the following
two equations:
.835x + .667y = .168,
.333x + .266y = .067.

(a) Use 5-digit arithmetic to compute the slopes of each of the lines,
and then use 6-digit arithmetic to do the same. In each case,
sketch the graphs on a coordinate system.
(b) Show by diagram why a small perturbation in either of these
lines can result in a large change in the solution.
(c) Describe in geometrical terms the situation that must exist in
order for a system to be optimally well-conditioned.

1.6.4. Using geometric considerations, rank the following three systems accord-
ing to their condition.
1.001x − y = .235, 1.001x − y = .235,
(a) (b)
x + .0001y = .765. x + .9999y = .765.
1.001x + y = .235,
(c)
x + .9999y = .765.

1.6.5. Determine the exact solution of the following system:

8x +5y +2z =15,
21x +19y +16z =56,
39x +48y +53z = 140.

Now change 15 to 14 in the first equation and again solve the system
with exact arithmetic. Is the system ill-conditioned?

1.6.6. Show that the system

v − w − x − y − z =0,
w − x − y − z =0,
x − y − z =0,
y − z =0,
z =1,