Ill-Conditioned Problem/Examples/Arbitrary Example 2
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Example of Ill-Conditioned Problem
Consider the simultaneous equations:
| \(\ds x - y\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds x - 1 \cdotp 0001 y\) | \(=\) | \(\ds 0\) |
These have the solution:
| \(\ds x\) | \(=\) | \(\ds 10 \, 001\) | ||||||||||||
| \(\ds y\) | \(=\) | \(\ds 10 \, 000\) |
However, the simultaneous equations:
| \(\ds x - y\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds x - 0 \cdotp 9999 y\) | \(=\) | \(\ds 0\) |
have the solution:
| \(\ds x\) | \(=\) | \(\ds -9999\) | ||||||||||||
| \(\ds y\) | \(=\) | \(\ds -10 \, 000\) |
So a change in the $4$th decimal place of one coefficient leads to a completely different solution.
This can be explained by the fact that the matrix of coefficients is nearly singular.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ill-conditioned
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ill-conditioned