Page 40 - Matrix Analysis & Applied Linear Algebra

32 Chapter 1 Linear Equations

(c) Using 3-digit arithmetic, column scale the coefficients by chang-
ing units: convert pounds of silica to tons of silica, pounds of
iron to half-tons of iron, and pounds of gold to troy ounces of
gold (1 lb. = 12 troy oz.).
(d) Use 3-digit arithmetic with partial pivoting to solve the column
scaled system of part (c). Then approximate the exact solution
by using your machine’s (or calculator’s) full precision with par-
tial pivoting to solve the system in part (c), and compare this
with your 3-digit solution by computing the relative error e r as
defined in part (b).

1.5.6. Consider the system given in Example 1.5.3.
(a) Use 3-digit arithmetic with partial pivoting but with no scaling
to solve the system.
(b) Now use partial pivoting with scaling. Does complete pivoting
provide an advantage over scaled partial pivoting in this case?

1.5.7. Consider the following well-scaled matrix:
 
1 0 0 ··· 0 0 1
 −1 1 0 ··· 0 0 1 
. . 
 

 −1 −1 1 . 0 0 1 
 . . . . . . . 
W n =  . . . . . . . . . .  .
 . . . . 
.
 
−1 −1 −1 1 0 1
 . . 
 
−1 −1 −1 ··· −1 1 1
 
−1 −1 −1 ··· −1 −11
(a) Reduce W n to an upper-triangular form using Gaussian elimi-
nation with partial pivoting, and determine the element of max-
imal magnitude that emerges during the elimination procedure.
(b) Now use complete pivoting and repeat part (a).
(c) Formulate a statement comparing the results of partial pivoting
with those of complete pivoting for W n , and describe the effect
this would have in determining the t -digit solution for a system
whose augmented matrix is [W n | b].
1.5.8. Suppose that A is an n × n matrix of real numbers that has been scaled
so that each entry satisfies |a ij |≤ 1, and consider reducing A to tri-
angular form using Gaussian elimination with partial pivoting. Demon-
strate that after k steps of the process, no entry can have a magnitude
k
that exceeds 2 . Note: The previous exercise shows that there are cases
where it is possible for some elements to actually attain the maximum
magnitude of 2 k after k steps.