Linear Algebra with Applications, 10th Edition

7.3 Pivoting Strategies

In this section, we present an algorithm for Gaussian elimination with row interchanges. At each step of the algorithm, it will be necessary to choose a pivotal row. We can often avoid unnecessarily large error accumulations by choosing the pivotal rows in a reasonable manner.

Gaussian Elimination with Interchanges

Consider the following example.

Example 1

Let

$$A=\left[\begin{array}{ccc}6& \hfill -4& 2\\ 4& \hfill 2& 1\\ 2& \hfill -1& 1\end{array}\right]$$

We wish to reduce A to triangular form by using row operations I and III. To keep track of the interchanges, we will use a row vector p. The coordinates of p will be denoted by $p\left(1\right),p\left(2\right)$, and $p\left(3\right)$. Initially, we set $\mathbf{p}=(1,2,3)$. Suppose that, at the first step of the reduction process, the third row is chosen as the pivotal row. Then instead ...