Numerical stability

Summary

An algorithm is numerically stable when the errors it makes, such as round-off errors, are not amplified beyond what the problem itself forces. The strongest common form is backward stability: the computed answer is the exact answer for a slightly perturbed input, as in backward error analysis. A stable algorithm applied to a well-conditioned problem gives an accurate answer; for an ill-conditioned problem even a stable algorithm can give large errors. For methods for differential equations, stability means instead that errors in the steps stay bounded, as checked by von Neumann stability analysis.