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Chapter 9

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.

Step-by-Step

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:
−9999x_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.