Ill-Conditioned Problem/Examples

Examples of Ill-Conditioned Problems

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.