Numerical Stability

In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is derived from the accuracy of the algorithm.

An opposite phenomenon is instability. Typically, algorithms would approach the right solution in the limit, if there were no round-off or truncation errors, but depending on the specific computational method, errors can be magnified, instead of damped, causing the error to grow exponentially.

Sometimes a single calculation can be achieved in several ways, all of which are algebraically equivalent in terms of ideal real or complex numbers, but in practice when performed on digital computers yield different results. Some calculations might damp out approximation errors that occur; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called numerically stable. One of the common tasks of numerical analysis is to try to select algorithms which are robust – that is to say, have good numerical stability among other desirable properties.