Civil Engineering Reference
In-Depth Information
If these equations are subtracted from the approximate equations, the
following is obtained:
+
…
+
axaxax axCCe
axaxax
∆
+
∆
+
∆
∆
=
− ′ =
11
1
12
2
13
3
1
n
n
1
1
1
+
…
+
axCCe
axaxax axCC
∆
+
∆
+
∆
∆
=
− ′ =
21
1
22
2
23
3
2
n
n
2
2
2
+
…
+
−
′
=
∆
+
∆
+
∆
∆
=
e
31
1
32
2
33
3
3
n
n
3
3
3
+
…
+
axaxa
n
∆
+
∆
+
∆
x
a xCCe
∆
=
− ′ =
1
1
n
2
2
n
3
3
nn
n
n
n
This shows that the corrections, ∆
x
′s, can be obtained by replacing the
constant vector of the solution with the difference of the constant vectors,
(
C
-
C′
s)'s, and applying reduction to find the error. These are then added
to the approximate solution and the process is repeated until accuracy is
achieved.
Example 2.12
Error equations
Determine the solution to the following set of equations using any
Gauss-Jordan elimination, but only carry two decimals of accuracy
(i.e.,
x.xx
) then apply error equations to increase accuracy.
211211
.
x
+
.
x
−
304111
.
x
+
.
x
=
165
.
1
2
3
4
−
002123
.
x
+
.
x
+
222102
.
x
+
.
x
=
13 18
.
1
2
3
4
014006
.
x
−
.
x
+
121108
.
x
−
.
x
= −
067
.
1
2
3
4
132020
.
x
+
.
x
+
0
.00
x
+
390 732
.
x
=
.
1
2
3
4
The process of three complete cycles is shown in Tables 2.11-2.13.
2.11
MAtRiX inVERSiOn MEtHOD
The solution to a set of linear equations can be achieved by using any reduc-
tion technique on the coefficient matrix augmented with the identity matrix.
aaa
a
100
0
11
12
13
1
n
aaa
a
010
0
21
22
23
2
n
[
]
=
AI
|
aaa
a
001
0
31
32
33
3
n
aaa
a
nn
000 1
n
1
n
2
n
3
From here, the coefficient matrix is reduced until the identity matrix is
on the left and the original identity on the right becomes the invert of A.
