
16.5 Root Finding 739
there are lots of matrix norms to choose from, there are many possible bounds. One
such bound is Cauchy’s bound,
|t|≤max {|a
0
|,1+|a
1
|,...,1+|a
n−1
|} = 1 + max{|a
0
|,...,|a
n−1
|}
Another bound that can be obtained is the Carmichael and Mason bound,
|t|≤
6
7
7
8
1 +
n−1
i=0
|a
i
|
2
If a
0
= 0, then f(0) = 0, so the roots of f are bounded away from zero. It is possible
to construct lower bounds by using g(t) = [t
n
f(1/t)]/a
0
. The roots of g(t) are the
reciprocal roots of f(t). Cauchy’s bound applied to g(t), then taking reciprocals, is
|t|≥
|a
0
|
1 + max{1, |a
1
|,...,|a
n−1
|}
The Carmichael and Mason bound is
|t|≥
|a
0
|
1 +
n−1
i=0
|a
i
|
2
These bounds are