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 alg(z); this includes the effects of roundoff. If where is small, then we say alg is a backward stable algorithm for f, or that the backward error is small. In other words, alg(z) is the exact value of f at a slightly perturbed input .
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
Thus, if is small, and the derivative is moderate, the error alg(z) - f(z) will be small. This is often written in the similar form
This approximately bounds the relative error by the product of the condition number of f at z, , and the relative backward error . Thus we get an error bound by multiplying a condition number and a backward error (or bounds for these quantities). We call a problem ill-conditioned if its condition number is large, and ill-posed if its condition number is infinite (or does not exist).
If f and z are vector quantities, then is a matrix (the Jacobian). So instead of using absolute values as before, we now measure by a vector norm and by a matrix norm . 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 described: when applied to a matrix A they produce the exact result for a slightly different matrix A + E, where is of order .
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 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 . 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.