Definition:Ill-Conditioned Problem
Definition
A problem in numerical analysis is ill-conditioned if and only if it is sensitive to small changes in the input data.
This sensitivity affects all methods of solution of the problem.
Hence such problems are hard to solve accurately.
Examples
Arbitrary Example $1$
Consider the zeroes of the polynomials:
| \(\ds \map p x\) | \(=\) | \(\ds x^8\) | ||||||||||||
| \(\ds \map q x\) | \(=\) | \(\ds x^8 - 10^8\) |
The zeroes of $\map p x$ are all $0$.
However, the zeroes of $\map q x$ are all of modulus $0 \cdotp 1$.
Hence, while $\map q x$ is a tiny perturbation of $\map p x$, the zeroes differ by a modulus some $8$ orders of magnitude larger.
Arbitrary Example $2$
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.
Also see
- Results about ill-conditioned problems can be found here.
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