`What happens with our method when applied to the system
`

`The unpleasant answer is that, although the system is well
defined with solution
,
our method simply fails. We try to
zero by subtracting
the first row, multiplied by a suitable constant , from the second row;
since , no matter
what constant we choose, the difference
.
`

`We can remedy things if we change the order of the equations, such that
. Obviously, changing the order of the equations
does not affect the solution. For example, interchanging
equations 1 and 3 means permuting rows 1 and 3 in , :
`

`Note that we have two distinct choices when making :
interchange rows 1 and 3, or interchange rows 1 and 2.
We will always choose the permutation that maximizes
the absolute value of . The reason is that this choice
minimizes the propagation
of roundoff errors during the computation (a rigorous proof of this fact
is beyond the scope of this class).
`

`In our example, interchanging rows
makes ,
while interchanging rows
makes ;
hence we will use
.
`

`The process of interchanging rows in order to maximize
is called partial pivoting (``partial'' because
the maximum is selected from
,
i.e. only from a ``part'' of the matrix).
`