The strategy of GE is to reduce a problem that we can't solve (a full linear
system) to one that we can (a triangular system), using elementary row
operations. There are n-1 stages, begining with 
,
, and finishing with the upper triangular system
.
At the kth stage we have converted the original system to 
, where
with 
upper triangular.
The purpose of the kth stage of the elimination is to zero the elements
below the diagonal in the kth column of 
. This is accomplished
by the operations
where the multipliers 
. At the
end of the 
st stage we have the upper triangular system
, which is solved by back substitution.
For an 
matrix, GE requires 
flops.
There are two problems with the method described above. They are:
.
is
large, then there is a possible loss of significance: in the subtraction
, low-order digits of 
could be lost.
Losing these digits could correspond to making a relatively large change to
the original matrix 
. The simplest example of this phenomenan is
for the matrix:
In this case, 
, and 
,
if 
(machine-precision), which would be the exact answer if
we changed
from 1 to 0. In this case, the matrix 
is
very well conditioned. The problem, of course, is with the choice of
as the pivot.
This problem can be solved by use of partial or complete pivoting. In partial pivoting, at the start of the kth stage, the kth and rth rows are interchanged, where
Partial pivoting ensures that the multipliers are nicely bounded:
A more expensive pivoting strategy, which interchanges both rows and columns is complete pivoting. At the start of the kth stage, rows k and r and columns k and s are inter-changed, where
This requires 
comparisons in total, compared with 
for
partial pivoting. Because of the searching overhead, and because partial
pivoting works so well, complete pivoting is rarely used in practice.