2.5. Vector Norms
29
Exercises
1. Relative Conditioning of Multiplication.
Let
x
,
y
,
˜
x
,
˜
y
be nonzero scalars. Show:
v
v
v
v
xy
− ˜
x
˜
y
xy
v
v
v
v
≤
(
2
+
ǫ)ǫ
,
where
ǫ
=
max
bv
v
v
v
x
− ˜
x
x
v
v
v
v
,
v
v
v
v
y
− ˜
y
y
v
v
v
v
B
,
and if
ǫ
≤
1, then
v
v
v
v
xy
− ˜
x
˜
y
xy
v
v
v
v
≤
3
ǫ
.
Therefore, if the relative error in the inputs is not too large, then the condition
number of multiplication is at most 3. We can conclude that multiplication is
wellconditioned in the relative sense, provided the inputs have small relative
perturbations.
2. Relative Conditioning of Division.
Let
x
,
y
,
˜
x
,
˜
y
be nonzero scalars, and let
ǫ
=
max
bv
v
v
v
x
− ˜
x
x
v
v
v
v
,
v
v
v
v
y
− ˜
y
y
v
v
v
v
B
.
Show: If
ǫ <
1, then
v
v
v
v
x/y
− ˜
x/
˜
y
x/y
v
v
v
v
≤
2
ǫ
1
−
ǫ
,
and if
ǫ <
1
/
2, then
v
v
v
v
x/y
− ˜
˜
y
x/y
v
v
v
v
≤
4
ǫ
.
Therefore, if the relative error in the operands is not too large, then the
condition number of division is at most 4. We can conclude that division is
wellconditioned in the relative sense, provided the inputs have small relative
perturbations.
2.5 Vector Norms
In the context of linear system solution, the error in the solution constitutes a
vector. If we do not want to pay attention to individual components of the error,
perhaps because there are too many components, then we can combine all errors
into a single number. This is akin to a grade point average which combines all
grades into a single number. Mathematically, this “combining” is accomplished
by norms. We start with vector norms, which measure the length of a vector.
Defnition 2.6.
A vector norm
b·b
is a function from
C
n
to
R
with three properties:
N1:
b
x
b ≥
0
for all
x
∈
C
n
, and
b
x
b =
0
if and only if
x
=
0
.
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2. Sensitivity, Errors, and Norms
N2:
b
x
+
y
b ≤ b
x
b+b
y
b
for all
x
,
y
∈
C
n
(triangle inequality).
N3:
b
α x
b = 
α
b
x
b
for all
α
∈
C
,
x
∈
C
n
.
The vector
p
norms below are useful for computational purposes, as well as
analysis.
Fact 2.7 (Vector
p
Norms).
Let
x
∈
C
n
with elements
x
=
(
x
1
...
x
n
)
T
. The
pnorm
b
x
b
p
=
n
s
j
=
1

x
j

p
1
/p
,
p
≥
1,
is a vector norm.
Example.
•
If
e
j
is a canonical vector, then
b
e
j
b
p
=
1 for
p
≥
1.
•
If
e
=
(
1
1
···
1
)
T
∈
R
n
, then
b
e
b
1
=
n
,
b
e
b
∞
=
1,
b
e
b
p
=
n
1
/p
,
1
< p <
∞
.
The three
p
norms below are the most popular, because they are easy to
compute.
•
One norm:
b
x
b
1
=
∑
n
j
=
1

x
j

.
•
Two (or Euclidean) norm:
b
x
b
2
=
r
∑
n
j
=
1

x
j

2
=
√
x
∗
x
.
•
InFnity (or maximum) norm:
b
x
b
∞
=
max
1
≤
j
≤
n

x
j

.
Example.
If
x
=
(
1
2
n
)
T
∈
R
n
, then
b
x
b
1
=
1
2
n(n
+
1
)
,
b
x
b
2
=
R
1
6
n(n
+
1
)(
2
n
+
1
)
,
b
x
b
∞
=
n
.
The inequalities below bound inner products in terms of norms.
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 Spring '07
 McClain
 Linear Algebra, Multiplication, Scalar, Norm, normed vector space

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