Civil Engineering Reference
In-Depth Information
+++=
+++=
−+−+=
+++
xxxx
x x xx
xxxx
xxxx
10
1
2
3
4
842
26
1
2
3
4
2
1
2
3
4
000
=
4
1
2
3
4
=
[]
T
C
A
[
A
−
1
]
a
nd
c
=−
()
1
ij
+
A
ij
ji
The determinants are shown by row expansion for the 4 × 4 matrix and by
the basket weave for all the 3 × 3 matrices in Table 2.3.
The last step in Table 2.3 is to multiply the invert of
A
, [
A
]
−1
, by the
constant vector, [
C
], to get the final solution vector, [
x
].
2.6
gAuSSiAn ELiMinAtiOn MEtHOD
This method is named for Carl Friedrich Gauss who developed it in 1670
(Newton 1707). It was referenced by the Chinese as early as 179 (Anon
179). Gaussian elimination is a method for solving matrix equations by
composing an augmented matrix, [
A
|
C
], and then utilizing elementary
row operations to reduce this matrix into upper triangular form, [
U
|
D
].
The elementary row operations used to reduce the matrix into upper trian-
gular form consist of multiplying an equation by a scalar, or adding two
equations to form another equation. The equation used to eliminate terms
in other equations is referred to as the
pivot equation
. The coefficient of
the pivot equation that lies in the column of terms to be eliminated is
called the
pivot coefficient
or
pivot element
.
If this coefficient of the pivot element is zero, the pivot row must be
exchanged with another row. This exchange is called
partial pivoting
. If the
row with the largest element in the pivot column is exchanged to the pivot
row, accuracy is increased. Partial pivoting is when the rows are interchanged
and full pivoting is when both the rows and columns are reordered to place
a particular element in the diagonal position prior to a particular operation.
Whenever partial pivoting is utilized, the determinant changes sign with
each pivot unless done before reduction starts. However, the value of the
determinant of the matrix is not affected by elementary row operations.
Once the matrix is reduced to an equivalent upper triangular matrix,
the solutions are found by solving equations by back substitution. The fol-
lowing is the reduction procedure in algorithmic form:
k
−
1
aa
a
a
k
+≤ ≤
+≤≤
1
1
j m
=−
(
)
kj
k
k
−
1
k
−
1
a
where
ij
ij
ik
k
−
1
k
i
n
kk
