Use Gauss elimination to solve 0.0003x1 + 3.0000x2 = 2.0001 1.0000x1 + 1.0000x2 = 1.0000 Note that in this form the first pivot element, a11 = 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 x1 = 1∕3 and x2

Question

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

Accepted Answer

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

Significant Figures	[latex]x_2[/latex]	[latex]x_1[/latex]	Absolute Value of Percent Relative Error for [latex]x_1[/latex]
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 [latex]x_1[/latex] is highly dependent on the number of significant figures. This is because in Eq. (E9.4.1), we are subtracting two almost-equal numbers.

[latex]x_1 =\frac{2.00001 −3( 2∕3)}{0.0003}[/latex] (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
[latex]1.0000x_1 + 1.0000x_2 = 1.0000[/latex]
[latex]0.0003x_1 + 3.0000x_2 = 2.0001[/latex]
Elimination and substitution again yields [latex]x_2 = 2∕3.[/latex] For different numbers of significant figures, [latex]x_1[/latex] can be computed from the first equation, as in
[latex]x_1 = \frac{1−(2∕3)}{1}[/latex]
This case is much less sensitive to the number of significant figures in the computation:

Significant Figures	[latex]x_2[/latex]	[latex]x_1[/latex]	Absolute Value of Percent Relative Error for [latex]x_1[/latex]
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.

Question 9.4: Use Gauss elimination to solve 0.0003x1 + 3.0000x2 = 2.0001 ...

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