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## Q. 9.4

Use Gauss elimination to solve
$0.0003x_1 + 3.0000x_2 = 2.0001$
$1.0000x_1 + 1.0000x_2 = 1.0000$
Note that in this form the first pivot element, $a_{11} = 0.0003,$ is very close to zero. Then repeat the computation, but partial pivot by reversing the order of the equations. The exact solution is $x_1 = 1∕3 and x_2 = 2∕3.$

## Verified Solution

Multiplying the first equation by 1∕(0.0003) yields
$x_1 + 10,000x_2 = 6667$
which can be used to eliminate $x_1$ from the second equation:
−9999$x_2$ = −6666
which can be solved for $x_2 = 2∕3.$ This result can be substituted back into the first equation to evaluate $x_1$:
$x_1 =\frac{2.00001 −3( 2∕3)}{0.0003}$                                                   (E9.4.1)
Due to subtractive cancellation, the result is very sensitive to the number of significant figures carried in the computation:

 Significant  Figures $x_2$ $x_1$ Absolute Value of  Percent Relative  Error for $x_1$ 3 0.667 −3.33 1099 4 0.6667 0.0000 100 5 0.66667 0.30000 10 6 0.666667 0.330000 1 7 0.6666667 0.3330000 0.1

Note how the solution for $x_1$ is highly dependent on the number of significant figures. This is because in Eq. (E9.4.1), we are subtracting two almost-equal numbers.

$x_1 =\frac{2.00001 −3( 2∕3)}{0.0003}$                                                   (E9.4.1)
On the other hand, if the equations are solved in reverse order, the row with the larger pivot element is normalized. The equations are
$1.0000x_1 + 1.0000x_2 = 1.0000$
$0.0003x_1 + 3.0000x_2 = 2.0001$
Elimination and substitution again yields $x_2 = 2∕3.$ For different numbers of significant figures, $x_1$ can be computed from the first equation, as in
$x_1 = \frac{1−(2∕3)}{1}$
This case is much less sensitive to the number of significant figures in the computation:

 Significant  Figures $x_2$ $x_1$ Absolute Value of  Percent Relative  Error for $x_1$ 3 0.667 0.333 0.1 4 0.6667 0.3333 0.01 5 0.66667 0.33333 0.001 6 0.666667 0.333333 0.0001 7 0.6666667 0.3333333 0.0000

Thus, a pivot strategy is much more satisfactory.