Page 11-35
While performing pivoting in a matrix elimination procedure, you can improve
the numerical solution even more by selecting as the pivot the element with the
largest absolute value in the column and row of interest. This operation may
require exchanging not only rows, but also columns, in some pivoting
operations. When row and column exchanges are allowed in pivoting, the
procedure is known as full pivoting
.
When exchanging rows and columns in partial or full pivoting, it is necessary to
keep track of the exchanges because the order of the unknowns in the solution
is altered by those exchanges. One way to keep track of column exchanges in
partial or full pivoting mode, is to create a permutation matrix
P = I
n×n
, at the
beginning of the procedure. Any row or column exchange required in the
augmented matrix A
aug
is also registered as a row or column exchange,
respectively, in the permutation matrix. When the solution is achieved, then,
we multiply the permutation matrix by the unknown vector x to obtain the order
of the unknowns in the solution. In other words, the final solution is given by
P⋅x = b’, where b’ is the last column of the augmented matrix after the solution
has been found.
Example of Gauss-Jordan elimination with full pivoting
Let’s illustrate full pivoting with an example. Solve the following system of
equations using full pivoting and the Gauss-Jordan elimination procedure:
X + 2Y + 3Z = 2,
2X + 3Z = -1,
8X +16Y- Z = 41.
The augmented matrix and the permutation matrix are as follows:
Store the augmented matrix in variable AAUG, then press ‚ @AAUG to get a
copy in the stack. We want to keep the CSWP (Column Swap) command
readily available, for which we use: ‚N~~cs~ (find CSWP),
@@OK@@. You’ll get an error message, press $, and ignore the message.
Next, get the ROW menu available by pressing: „Ø @)CREAT @)@ROW@.
.
100
010
001
,
411168
1302
2321
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−=
PA
aug
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