Round-off Error for Gaussian Elimination

Let us say we have are solving the following system with Gaussian elimination and can only carry 3 significant digits:

$$\begin{array}{rl}2x_1+100,000x_2& =100,000\\ x_1+x_2& =2\end{array}$$

Forward elimination gives:

$$\begin{array}{rl}2x_1+100,000x_2& =100,000\\ -50,000x_2& =-50,000\end{array}$$

which would give $x_1=0.0$ ad $x_2=1.0$, an answer that is obviously not correct.

However, if we didn’t round off, this equation would be:

$$\begin{array}{rl}-49,999x_2& =-49,998\\ x_2& =0.99998\\ x_1& =1.00002\end{array}$$

If we use Scaling and Pivoting, we instead get:

$$\begin{array}{rl}{x}_1+x_2& =2\\ 0.00002x_1+x_2& =1\end{array}$$

which gives us $x_1=1.00$ and $x_2=1.00$, an answer that is accurate to 3 significant figures.