Unformatted text preview: to the largest absolute
˜
error,
x − x ∞ = max xj − xj .
˜
˜
1 ≤j ≤n For the one and two norms we have
max xj − xj  ≤ x − x
˜
˜ 1 1≤j ≤n ≤ n max xj − xj 
˜
1 ≤j ≤n and
max xj − xj  ≤ x − x
˜
˜ 1≤j ≤n 2 ≤ √ n max xj − xj .
˜
1≤j ≤n Hence absolute errors in the one and two norms can overestimate the worst componentwise error by a factor that depends on the vector length n.
Unfortunately, normwise relative errors give much less information about
componentwise relative errors.
Example. Let x be an approximation to a vector x where
˜
x= 1
,
ǫ The normwise relative error 0 < ǫ ≪ 1, x −x ∞
˜
x∞ x=
˜ 1
.
0 = ǫ is small. However, the componentwise −˜
relative error in the second component, x2x2x2  = 1, shows that x2 is a totally
˜

inaccurate approximation to x2 in the relative sense.
The preceding example illustrates that a normwise relative error can be
small, even if individual vector elements have a large relative error. In the inﬁnity norm, for example, the normwise relative error only bounds the relative Downloaded 01/17/14 to 143.215.200.123. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 32 2. Sensitivity, Errors, and Norms error corresponding to a component of x with the largest magnitude. To see this,
let xk  = x ∞ . Then
˜
max1≤j ≤n xj − xj  xk − xk 
˜
x −x ∞
˜
≥
.
=
x∞
xk 
xk 
For the normwise relative errors in the one and two norms we incur additional
factors that depend on the vector length n,
x −x
˜
x1 1 ≥ ˜
1 xk − xk 
,
n xk  x −x
˜
x2 2 ˜
1 xk − xk 
.
≥√
n xk  Therefore, normwise relative errors give no information about relative errors in
components of smaller magnitude. If relative errors in individual vector components are important, then do not use normwise errors.
Remark 2.11. When measuring the normwise relative error of an approximation
˜
˜
x to x , the question is which error to measure, x −x or x −x ? If x ≈ x ,
˜
˜
x
x
˜
then the two...
View
Full
Document
 Spring '07
 McClain
 Linear Algebra, Multiplication, Scalar, Norm, normed vector space

Click to edit the document details