Page 38 - Matrix Analysis & Applied Linear Algebra

30 Chapter 1 Linear Equations
Exercises for section 1.5

1.5.1. Consider the following system:

10 −3 x − y =1,
x + y =0.

(a) Use 3-digit arithmetic with no pivoting to solve this system.
(b) Find a system that is exactly satisfied by your solution from
part (a), and note how close this system is to the original system.
(c) Now use partial pivoting and 3-digit arithmetic to solve the
original system.
(d) Find a system that is exactly satisfied by your solution from
part (c), and note how close this system is to the original system.
(e) Use exact arithmetic to obtain the solution to the original sys-
tem, and compare the exact solution with the results of parts (a)
and (c).
(f) Round the exact solution to three significant digits, and compare
the result with those of parts (a) and (c).

1.5.2. Consider the following system:

x + y =3,
5
5
−10x +10 y =10 .
(a) Use 4-digit arithmetic with partial pivoting and no scaling to
compute a solution.
(b) Use 4-digit arithmetic with complete pivoting and no scaling to
compute a solution of the original system.
(c) This time, row scale the original system first, and then apply
partial pivoting with 4-digit arithmetic to compute a solution.
(d) Now determine the exact solution, and compare it with the re-
sults of parts (a), (b), and (c).

1.5.3. With no scaling, compute the 3-digit solution of

−3x + y = −2,
10x − 3y =7,

without partial pivoting and with partial pivoting. Compare your results
with the exact solution.