Civil Engineering Reference
In-Depth Information
10
xx
+++=
+++=
+++=
+++ =
2
xx
50
1
2
3
4
2 0
x
x
x
2
x
63
1
2
3
4
x
2 0
x
xx
67
1
2
3
4
2
xx
x
10
x
75
1
2
3
4
Note that this is a diagonal system with 10's on the diagonal and all other
coefficients are much less. Begin the iteration by setting
x
1
= x
2
= x
3
= x
4
=
0
and solving for each of the unknowns using the corresponding equation in
a top down order.
++
()
+=∴=
10 020050
x
x
5 000
.
1
1
(
)
+++
()
=∴=
25000
.
10 020 3
x
x
5 300
.
2
2
+
(
)
++=∴=
5
000 25300
.
10 0 7
x
x
x
==∴=
5 140
.
3
3
(
)
+
25000
.
5 300 5 140 10
.
+
.
+
75
x
5 456
.
4
After completing the first cycle, start with the first equation using the new
values and find a closer approximation for each unknown. Also, check the
difference between the new values and the previous values to determine if
the desired accuracy is achieved.
+
(
)
+
10
x
+
5 300 25140
.
.
5 456
.
= ∴=
50
x
2 896
.
and ∆
x
=−
2 104
.
1
1
1
(
)
+
+
(
)
=∴=
22896
.
10
x
+
5 140 25456
.
.
63
x
4 116
.
and ∆
x
= −
1 184
.
2
2
2
(
)
++=∴=
2 896 24
.
+
116
10
x
5 456
.
67
x
5 042
.
and ∆
x
= −
0 098
.
3
3
3
(
)
+
22896 411
.
.
665042 10
+
.
+ =∴=
x
75
x
6 005
.
and ∆
x
=
0 549
.
4
4
4
None of the values of ∆
x
are less than
e
= 0.01, so the process is repeated.
Table 2.18 shows the entire process to convergence. The process can be
stopped when each value has changed less than
e
or when a cycle results
in each value changing less than
e
.
2.13
EigEnVALuES bY cRAMER'S RuLE
A homogeneous equation is one where all the constants on the right-hand
side of the equal sign are zero. The typical set of
n
homogeneous equations
with
n
unknown solution sets is as follows:
