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).