Civil Engineering Reference
In-Depth Information
2.2 Definition.
Let
A
be a symmetric positive definite
N
×
N
matrix, and suppose
s
∈ R
. Then
=
(x, A
s
x)
1
/
2
|||
x
|||
s
:
(
2
.
5
)
N
.
defines a norm, where
(
·
,
·
)
is the Euclidean scalar product in
R
Using (2.3), (2.4), and the orthogonality relation, we have
(x, A
s
x)
=
k
c
k
z
k
,
i
c
i
λ
i
z
i
=
λ
i
c
i
.
i
Thus the norm (2.5) has the following alternative representation:
N
|||
x
|||
s
=
2
1
/
2
λ
i
|
c
i
|
=
A
s/
2
x
.
(
2
.
6
)
i
=
1
2.3 Properties of the Norm (2.5).
(1)
Connection with the Euclidean norm:
|||
x
|||
=
x
, where
·
is the Euclidean
0
norm.
(2)
Logarithmic convexity:
For
r, t
1
∈ R
and
s
=
2
(r
+
t)
,
1
/
2
t
1
/
2
r
|||
x
|||
s
≤ |||
x
|||
· |||
x
|||
,
|
(x, A
s
y)
| ≤ |||
x
|||
r
· |||
y
|||
t
.
Indeed, with the help of the Cauchy-Schwarz inequality, it follows that
|
(x, A
s
y)
|=|
(A
r/
2
x,A
t/
2
y)
|≤
A
r/
2
x
A
t/
2
y
= |||
x
|||
r
|||
y
|||
t
.
This is the second inequality. The first follows if we choose
x
=
y
. Taking the
logarithm of both sides and using the continuity, we see that the function
s
−→
log
|||
x
|||
s
is convex provided that
x
=
0.
(3)
Monotonicity:
Let
α
be the constant of ellipticity, i.e.,
(x, Ax)
≥
α(x,x)
. Then
α
−
t/
2
|||
x
|||
t
≥
α
−
s/
2
|||
x
|||
s
,
for
t
≥
s.
1.
Otherwise we have the monotonicity property for the normalized matrix
α
−
1
A
which implies the monotonicity as stated for
A
.
(4)
Shift theorem.
The solution of
Ax
=
b
satisfies
For the special case
α
=
1, this follows immediately from (2.6) and
λ
i
≥
α
=
|||
x
|||
s
+
2
= |||
b
|||
s
. This follows from
(x, A
s
+
2
x)
(Ax, A
s
Ax)
(b, A
s
b).
for all
s
∈ R
=
=
Using the scale defined with (2.5), we immediately get the following property
of Richardson relaxation without any additional hypotheses. It can be thought of
as a smoothing property, as we shall see later.
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