## 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.