| The standard technique for detecting instability in numerical algorithms is backward error analysis. The analysis is difficult and tedious when performed by hand, while attempts to automate it have always placed severe restrictions on the tested numerical algorithms. A new approach for detecting instability, functional stability analysis, removes these restrictions by treating numerical algorithms as black boxes. The approach consists of two parts. The first part uses the relationship between the forward error, the backward error, and a problem's condition to define a function that estimates a lower bound on the backward error. In the second part, a new optimization method maximizes the function. A numerical algorithm is unstable if the maximization shows that the backward error can become large. Since numerical algorithms are treated as black boxes, functional stability analysis normally requires little more than an executable version of a numerical algorithm to determine if it is unstable. |
