Civil Engineering Reference
In-Depth Information
[[]
=
[] [[]
−
[]
=
Ax C
or
A xC
0
If the original system of equations is reduced into an equivalent system in
upper triangular form, the following is true:
1
01
00 1
uu
u
x
x
x
d
d
d
12
13
1
1
=
23
2
2
3
3
[[]
−
[]
= 0
UUx D
Also, a lower triangular matrix exists, such that, when the first set is
pre-multiplied by [
L
], the result is the second set as follows:
l
l
00
0
11
l
21
22
l
l
l
31
32
33
(
)
=
[[]
−
[]
[][[]
−
[]
LUxD Ax C
[[][]
=
[] [][]
=
[]
L
UA LD C
and
In matrix form, it looks as follows:
l
l
00
0
1
01
00 1
uud
ud
d
aaac
aaac
aaac
11
12
13
1
11
12
13
1
l
=
21
22
23
2
21
22
23
2
l
l
l
31
32
33
3
31
32
33
3
The order of the solution process is as follows with each producing an
equation involving only one unknown:
1. Obtain column 1 of [
L
] by multiplying each row of [
L
] by column 1
of [
U
] to get column 1 of [
A
]. That is, use
a
11
,
a
21
,
a
31
to get
l
11
,
l
21
,
l
31
.
2. Obtain row 1 of [
U
] by multiplying row 1 of [
L
] times each column
of [
U
] to get row 1 of [
A
], excluding column 1 of [
U
]. That is, use
a
12
,
a
13
,
c
1
to get
u
12
,
u
13
,
d
1
.
