Civil Engineering Reference
In-Depth Information
The simulated time series are shown in Fig. A.5 (
T
=
600
s
and
Δ =
0.06
s). The standard
deviation of the process as calculated from the parent spectrum is
. The standard
deviations of the three simulated time series are 2.414, 2.328 and 2.3995. The discrepancy (less
than about 3 %) is caused by the unduly coarse frequency segmentation.
σ =
2.3365
x
A.4 The Cholesky decomposition
Given a positive definite and symmetric matrix
X
, the Cholesky decomposition of
X
is
defined by a lower triangular matrix
Y
of the same size that satisfies the following:
XYY
(A.24)
T
=
Expanding this equation
x
x
x
y
0
0
y
y
y
ª
""
º ª
º ª
º
11
1
i
1
N
11
11
1
i
1
N
«
» «
» «
»
#
# #
""
«
» «
» «
»
«
x
x
x
» «
y
y
0
» «
0
y
y
»
=
⋅
i
1
ii
iN
i
1
ii
ii
iN
«
» «
» «
»
#
# #
""
«
» «
» «
»
«
» «
» «
»
x
x
x
y
y
y
0
0
y
¬
¼ ¬
¼
¬
¼
N
1
Ni
NN
N
1
Ni
NN
NN
(A.25)
and developing the matrix multiplication column by column, it is seen that the first
column renders
xyy
y x
xyy
y xy
-
-
=⋅
½
=
11
11
11
11
11
°
°
°
=⋅
/
=
°
° °
21
21
11
21
21
11
(A.26)
®
¾ ®
#
#
°
° °
°
° °
x
y
y
=⋅
y xy
/
=
¯
¿
¯
N
1
N
1
11
N
1
N
1
11
while the second column renders
xyyyy y x y y
xyyyy y xyyy
=⋅ +⋅
-
-
½
=
−
22
21
21
22
22
22
22
21
21
°
°
°
(
)
=⋅ +⋅
/
=−
°
° °
32
31
21
32
22
32
32
31
21
22
(A.27)
®
¾ ®
#
#
°
° °
°
° °
(
)
x
=⋅ +⋅
y
y
y
y
y
x
y y
/
y
=
−
¯
¿
¯
N
2
N
1
21
N
2
22
N
2
N
2
N
1
21
22
