Conditioning, Stability and Floating-Point Error
Conditioning says how much the answer moves when the data moves, and stability says how much an algorithm adds on top of that. Rounding error is tiny, and those two amplifiers are what make it visible.
What a learner can do afterwards
- Compute a condition number and read it as an error amplifier
- Tell an ill-conditioned problem apart from an unstable algorithm
- Show catastrophic cancellation on a concrete subtraction
1 · Read
Every stored number is rounded, and floating point rations about fifteen decimal digits. Most fractions refuse exact binary form, so 0.1 arrives already approximated. Adding a tiny number to a huge one may change nothing at all.
Conditioning says how the answer moves when the data moves. The condition number multiplies the norm of the matrix by the norm of its inverse, and it reads as an error amplifier. With norms 4 and 0.5 the number is 2, so input error at most doubles; a nearly singular system loses digits even under a stable algorithm.
Stability says how much the algorithm adds on top of that problem sensitivity. An ill-conditioned problem swings wildly by itself, while an unstable algorithm ruins even calm problems. A stable method on an ill-conditioned problem can still return a poor answer, and that is nobody's coding bug.
Subtracting nearly equal quantities exposes the rounding dust, which is called catastrophic cancellation. With 4-digit arithmetic, 1.0001 minus 1.0000 is the dangerous pair: the leading digits cancel and the result lives on leftovers. Pairs far apart are safe, and extra digits help only after the algebra is reworked.
Rounding is tiny, the condition number amplifies it, stability adds its own share, and cancellation of near twins is the classic trap.
More to read
1.5 Number basics
Every stored number is rounded, and floating point rations about fifteen decimal digits. Most fractions cannot be written exactly in binary, so 0.1 arrives already approximated. Subtracting nearly equal quantities then exposes the rounding dust.
2.4 Floating-point errors
Floating-point errors are not random noise; they follow the arithmetic operation by operation. Adding a tiny number to a huge one may change nothing, and innocent subtractions can detonate. The famous examples all share one shape: information destroyed before the final rounding. Studying them teaches where extra precision actually helps and where algebra must change.
1.3 Accuracy, Precision, and Significant Figures
Scientists report digits they trust, dropping the rest as significant figures. Accuracy says how close to truth; precision says how finely resolved. A condition number translates between them: it tells how many input digits survive to the output. Measurement language is therefore error analysis in disguise.
2 · Watch
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Where it sits
Learn first
Nothing; this is a starting point.
Optional connections
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.