numerical stability

Two cooks follow the same recipe (the same problem) but use different techniques (different algorithms). One technique magnifies every small slip in measuring into a ruined dish; the other stays forgiving and produces a fine result even with shaky hands. Numerical stability is the quality of the TECHNIQUE — the algorithm — that keeps the inevitable little rounding errors of floating-point arithmetic from snowballing into a wrecked answer.

Stability is a property of the algorithm, in deliberate contrast to conditioning, which is a property of the problem. The gold-standard form is backward stability: the computed answer is the exact answer to a problem within rounding distance of the one posed. A weaker but still useful form is mixed forward-backward stability: the computed answer is nearly the exact answer to a nearby problem. The unifying idea is that a stable algorithm does not amplify rounding errors beyond what the problem's own conditioning already forces; it adds essentially nothing of its own.

The crucial honest point: stability is necessary but not sufficient for an accurate answer. A perfectly stable algorithm still returns a poor result on an ill-conditioned problem, because accuracy equals conditioning times stability and the conditioning factor can be enormous. Conversely, an unstable algorithm can ruin an easy, well-conditioned problem all by itself. So you want a stable algorithm to ensure you are not making things worse — but you must check the conditioning separately to know whether the answer can be trusted.