Civil Engineering Reference
In-Depth Information
11111
16 8421
11111
81 27 931
16 8421
8
44
2
44
8
AC
=
−
−
−
−
−
−
From partial pivoting, the first row can be swapped with the fourth row to
form the following matrix:
81 27 931
16 8421
11111
11111
16 8421
−
−
44
44
2
8
8
AC
=
−
−
−
−
Now elimination may be performed as shown in Table 2.6. Note that for
each column reduction, elements are reduced to zero below and above
the pivot position. Once it reduces to a diagonal matrix, the solution is
found by dividing each row by the pivot element. The determinant is
found as |
A
| = (81)(13.3333)(1)(1.3333) (−2) = −2880.
2.9
cHOLESKY DEcOMPOSitiOn MEtHOD
Cholesky decomposition is also known as Crout's method or matrix
factorization. This method was discovered by André-Louis Cholesky
(
Commandant Benoit 1924).
Cholesky decomposition changes the orig-
inal augmented equation to an equivalent upper and lower triangular set.
If a set of three simultaneous equations exist, they can be represented as
follows:
aaa
aaa
aaa
x
x
x
c
c
11
12
13
1
1
=
21
22
23
2
2
c
3
31
32
33
3
If [
A
] represents the coefficient matrix, [
x
] represents the column matrix of
the unknowns, and [
C
] represents the column matrix of the constants, the
previous can be expressed as the following equation:
