Civil Engineering Reference
In-Depth Information
2.7
gAuSS-JORDAn ELiMinAtiOn MEtHOD
The Gauss-Jordan elimination method is a variation of the Gaussian
elimination method in which an unknown is eliminated from all equa-
tions except the pivot during the elimination process. This method was
described by Wilhelm Jordan in 1887 (Clasen 1888). When an unknown is
eliminated, it is eliminated from equations preceding the pivot equation as
well as those following the pivot equation. The result is a diagonal matrix
and eliminates the need for back substitution. In fact, if the pivot elements
are changed to “ones” by dividing each row by the pivot element, the last
column will contain the solution.
The disadvantage to the Gauss-Jordan elimination method is that two
matrices are required for elimination, however, they do get smaller as the
process continues. The previous pivot is moved to the bottom, with a new
pivot on top. The Gauss-Jordan process is shown in Example 2.9.
Example 2.9
Gauss-Jordan elimination method
Determine the solution to the following set of equations using Gauss-
Jordan elimination.
225 3
1
xxx
−+=
(2.1a)
2
3
(2.1b)
234 0
1
xx x
++=
2
3
(2.1c)
3
xx x
−+ =
3
10
1
2
3
The first step is to divide the first equation of the set by the coefficient of
the first unknown in that equation, 2. Equation 2.2a is then multiplied by
the corresponding coefficient of that unknown of Equations 2.1b and 2.1c
to give the following:
5
2
13
2
xx
−+ =
x
(2.2a)
1
2
3
225 3
1
xxx
−+=
(2.2b)
2
3
33
15
2
39
2
xx
−+ =
x
(2.2c)
1
2
3
