A numerical algorithm can introduce errors even when the underlying mathematical problem is well-conditioned. Numerical stability asks whether those computational errors are controlled rather than amplified unnecessarily.

A useful viewpoint is backward error. Instead of asking only how far the computed answer $\tilde y$ is from the exact answer, ask whether $\tilde y$ is the exact solution to a nearby input problem.

An algorithm is backward stable when its computed result can be interpreted as the exact answer for inputs perturbed only slightly from those supplied.

This separates two sources of difficulty:

• conditioning describes how sensitive the problem is;
• stability describes how faithfully the algorithm handles that problem.

A stable algorithm applied to an ill-conditioned problem can still have a large forward error, because the mathematical problem itself amplifies small perturbations.