Condition Number
A measure of how sensitive a problem's output is to small changes in its input.
Definition
The condition number quantifies how much a problem amplifies errors: it measures the maximum ratio of relative change in the output to relative change in the input. A problem with a large condition number is ill-conditioned, meaning small input errors can cause large output errors.
Ill-conditioning is not always fatal: it warns that a problem is sensitive, but reformulating it, by scaling variables or choosing better coordinates, can improve conditioning dramatically. Recognizing the warning is the first step toward a numerically sound formulation.
The condition number is a property of the problem, not the algorithm, and it sets a hard limit on achievable accuracy in finite precision: an ill-conditioned problem amplifies input and rounding errors regardless of how carefully it is solved. Recognizing ill-conditioning prevents trusting spurious digits, and often a reformulation, rescaling variables or choosing better coordinates, improves conditioning dramatically and restores a numerically sound problem.
Interpretation
- A condition number near 1 is well-conditioned and stable.
- A large condition number warns that results may be unreliable.
- For a linear system, it bounds how rounding error propagates.
Why it matters
Conditioning is a property of the problem itself, separate from the algorithm. Even a perfect algorithm cannot produce an accurate answer to an ill-conditioned problem in finite precision. Recognizing ill-conditioning prevents trusting spurious results.
Fusion connection
Monitoring condition numbers in Kronos linear systems flags when a simulation setup is numerically fragile, prompting reformulation before unreliable results propagate into design decisions.