Suppose that an approximation to is computed in an arithmetic of precision , where f is a real scalar function of a real scalar variable. How should we measure the ``quality" of ?
In many cases, we would be happy with a tiny relative error, , but this cannot always be achieved. Instead of focusing on the relative error of we can ask ``for what set of data have we actually solved our problem?", that is, for what do we have ? In general, there may be many such , so we should ask for the smallest one. The value of , possibly divided by |x|, is called the backward error. The absolute and relative errors of are called forward errors. Figure 3 highlights the relationship between these errors.
The process of bounding the backward error of a computed solution is called backward error analysis. It interprets rounding errors as being equivalent to perturbations in the data. The data frequently contains uncertainties due to previous computations or errors committed in storing numbers on the computer. If the backward error is no larger than these uncertainties then the computed solution can hardly be criticized.
A method for computing is called backward stable if, for any x, it produces a computed with a small backward error, that is, for some small . The definition of ``small" is context dependent. Many a times we use , where is the machine epsilon.