We illustrate standard error analysis with the simple example of evaluating the scalar function y=f(z). Let the output of the subroutine which implements f(z) be denoted ; this includes the effects of roundoff. If where is small, then we say is a backward stable algorithm for f, or that the backward error is small. In other words, is the exact value of f at a slightly perturbed input .4.5
Suppose now that f is a smooth function, so that
we may approximate it near z by a straight line:
Then we have the simple error estimate
If f and z are vector quantities, then f'(z) is a matrix (the Jacobian). So instead of using absolute values as before, we now measure by a vector norm and f'(z) by a matrix norm |f'(z)|. The conventional (and coarsest) error analysis uses a norm such as the infinity norm. We therefore call this normwise backward stability. For example, a normwise stable method for solving a system of linear equations Ax=b will produce a solution satisfying where and are both small (close to machine epsilon). In this case the condition number is (see section 4.4 below).
Almost all of the algorithms in LAPACK (as well as LINPACK and EISPACK) are stable in the sense just described4.8: when applied to a matrix A they produce the exact result for a slightly different matrix A+E, where is of order . In a certain sense, a user can hardly ask for more, provided the data is at all uncertain.
It is often possible to compute the norm |E| of the actual backward error by computing a residual r, such as r=Ax-b or , and suitably scaling its norm |r|. The expert driver routines for solving Ax=b do this, for example. For details see [55,67,85,95].
Condition numbers may be expensive to compute exactly. For example, it costs about operations to solve Ax=b for a general matrix A, and computing exactly costs an additional operations, or twice as much. But can be estimated in only O(n2) operations beyond those necessary for solution, a tiny extra cost. Therefore, most of LAPACK's condition numbers and error bounds are based on estimated condition numbers, using the method of [59,62,63]. The price one pays for using an estimated rather than an exact condition number is occasional (but very rare) underestimates of the true error; years of experience attest to the reliability of our estimators, although examples where they badly underestimate the error can be constructed . Note that once a condition estimate is large enough, (usually ), it confirms that the computed answer may be completely inaccurate, and so the exact magnitude of the condition estimate conveys little information.