Ill-Conditioned Problem/Examples/Arbitrary Example 1

Example of Ill-Conditioned Problem

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.

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