Civil Engineering Reference
In-Depth Information
+
…
+
ax ax ax
+
+
ax C
=
11 1
122 13
3
1
nn
1
+
…
+
ax ax ax ax C
+
+
=
21 1
222 23
3
2
nn
2
+
…
+
a
xaxax
+
+
a xC
nn
=
3
111 32
2
333
3
3
+
…
+
ax ax ax ax C
n
+
+
=
11 22 33
n
n
nnn n
If the approximate roots
x
′
1
,
x
′
2
,
x
′
3
…,
x
′
n
have been obtained by elimina-
tion, upon substitution into the equations the constants
C′
1
,
C′
2
,
C′
3
…,
C′
n
are found as follows:
ax ax ax
′ +
′ +
′ +…+
ax C
′ = ′
11 1
122 13 3
1
nn
1
ax ax ax
′ +
′ +
′ +…+
a
xC
′ = ′
21 1
222 23 3
2
nnn
2
ax ax ax
′ +
′ +
′ +…+
ax C
′ = ′
31 1
322 33 3
3
nn
3
ax ax ax
′ +
′ +
′ +…+
ax C
′ = ′
n
11 22 33
n
n
nnn n
If ∆
x
1
, ∆
x
2
, ∆
x
3
…, ∆
x
n
are the corrections that must be added to the
approximate root to obtain the exact root values
x
1
,
x
2
,
x
3
,…,
x
n
, the
following is utilized:
xx x
=
′
+
∆
1
1
1
xx x
= ′ +
∆
2
2
2
xx x
= ′ +
∆
3
3
3
xx x
n
= ′ +
∆
n
n
If we substitute these expressions for the exact root, we obtain the following:
(
)
+
(
)
+
(
)
+
…
+
(
)
=
ax xax
′ +
∆
′ +
∆
x ax x
′ +
∆
a
x
′ +
∆
x
C
11
1
1
12
2
2
13
3
3
1
n
n
n
1
(
)
+
(
)
+
(
)
+
…
+
(
)
=
a
x
′ +
∆
x ax xax
′ +
∆
′ +
∆
x ax x C
′ +
∆
2
111 1
22
2
2
23
3
3
2
n
n
n
2
(
)
+
(
)
+
(
)
+
…
+
(
)
=
′′
+
′
+
′
+
′
+
a
x
∆
x ax xax
∆
∆
x ax x C
n
∆
31
1
1
32
2
2
33
3
3
3
n
n
3
(
)
+
(
)
+
(
)
+
…
+
(
∆
)
=
C
n
′
+
′
+
′
+
′
+
ax xax
∆
∆
x ax x
∆
a
x
x
n
11 1
n
2
2
2
n
3
3
3
nn
n
n
