12.2 Partial Pivoting

Partial pivoting is defined as interchanging the first row of the submatrix currently being looked at with a subsequent one. Its purpose is to replace the current pivot by a more desirable one. In particular, partial pivoting must be used if otherwise the pivot would be zero.

More generally, partial pivoting must be used to select the equation with the most desirable pivot. In particular

• In a numerical solution of systems of equations on a computer, the main objective is to minimize the effect of round-off errors. Here you should select the row having the pivot with the largest absolute value. (This assumes that the equations are scaled in a consistent, reasonable way.)
• In hand computations with integer matrices in homework or an exam in this class, the objective is to avoid fractions in the algebra. So you try to select a pivot which is $\pm1$.

  Note that there may not be such a pivot. In that case, you may be still be able to avoid fractions by partial pivoting. Or else you might be able to avoid them by multiplying the row being replaced by an integer greater than 1 instead of 1.

  If that works, it should normally be done. However, if you want to identify matrices $L$ and $U$ in the $LU$ theorem, you can only multiply the row being replaced by 1. Otherwise the theorem does not work. So you may have to live with fractions.