Civil Engineering Reference
In-Depth Information
3. Obtain column 2 of [
L
] by multiplying each row of [
L
] times column
2 of [
U
] to get column 2 of [
A
], excluding row 1 of [
L
], That is, use
a
22
,
a
32
to get
l
22
,
l
32
.
4. Obtain row 2 of [
U
] by multiplying row 2 of [
L
] times each column
of [
U
] to get row 2 of [
A
], excluding columns 1 and 2 of [
U
]. That
is, use
a
23
,
c
2
to get
u
23
,
d
2
.
5.
Obtain column 3 of [
L
] by multiplying each row of [
L
] times col-
umn 3 of [
U
] to get column 3 of [
A
], excluding row 1 and 2 of [
L
].
That is, use
a
33
to get
l
33
.
6.
Obtain row 3 of [
U
] by multiplying row 3 of [
L
] times each column
of [
U
] to get row 3 of [
A
], excluding columns 1, 2, and 3 of [
U
].
That is, use
c
3
to get
d
3
.
All of these arithmetic operations can be done with an algorithm as
follows:
l a
i
= …
=
123
1
234
,,,
,
n
=
For
i
1
i
1
j
a
a
j
= …+
=
,,,
,
n
1
1
j
u
=
For
1
j
i
1
11
= …
=++…
j
234
12
,,,
,
n
−
∑
j
1
l a lu
=−
For
i
, ,
n
foreachvalue of j
j
,
j
,
j
,
ij
,
ij
ik
kj
k
=
1
(
)
= …
=++…+
i
234
12 1
,,,
,
n
∑
1
i
−
a
−
l u
ij
ik
kj
k
=
u
=
For
jii
,
,
i
,
,
n
ij
,
l
(
)
ii
f
or each valueofi
xu
=
n
n n
,
+
1
n
∑
{
}
xu ux
=
−
For
in n
=− −−…
1231
,
,
n
,
,
i
i in
,
+
1
ij
j
ji
=+
1
An improved Cholesky decomposition scheme can also be used with only
one matrix in the process as follows:
a
a
1
j
{
}
a
=
For
j
= …+
234
,,,
,
n
1
1
j
11
