Improved Error Bounds

The standard error analysis just outlined has a drawback: by using the infinity norm $\Vert \delta \Vert _{\infty}$ to measure the backward error, entries of equal magnitude in $\delta$ contribute equally to the final error bound $\kappa (f,z) (\Vert \delta \Vert/\Vert z\Vert)$. This means that if z is sparse or has some very tiny entries, a normwise backward stable algorithm may make very large changes in these entries compared to their original values. If these tiny values are known accurately by the user, these errors may be unacceptable, or the error bounds may be unacceptably large.

For example, consider solving a diagonal system of linear equations Ax=b. Each component of the solution is computed accurately by Gaussian elimination: x_i = b_i / a_ii. The usual error bound is approximately $\epsilon \cdot \kappa_{\infty}(A) = \epsilon \cdot \max_i \vert a_{ii}\vert / \min_i \vert a_{ii}\vert$, which can arbitrarily overestimate the true error, $\epsilon$, if at least one a_ii is tiny and another one is large.

LAPACK addresses this inadequacy by providing some algorithms whose backward error $\delta$ is a tiny relative change in each component of z: $\vert \delta_{i} \vert = O( \epsilon ) \vert z_{i} \vert$. This backward error retains both the sparsity structure of z as well as the information in tiny entries. These algorithms are therefore called componentwise relatively backward stable. Furthermore, computed error bounds reflect this stronger form of backward error^4.9.

If the input data has independent uncertainty in each component, each component must have at least a small relative uncertainty, since each is a floating-point number. In this case, the extra uncertainty contributed by the algorithm is not much worse than the uncertainty in the input data, so one could say the answer provided by a componentwise relatively backward stable algorithm is as accurate as the data warrants [1].

When solving Ax=b using expert driver xyySVX or computational routine xyyRFS, for example, we almost always compute $\hat{x}$ satisfying $(A+E)\hat{x}=b+f$, where e_ij is a small relative change in a_ij and f_k is a small relative change in b_k. In particular, if A is diagonal, the corresponding error bound is always tiny, as one would expect (see the next section).

LAPACK can achieve this accuracy for linear equation solving, the bidiagonal singular value decomposition, and the symmetric tridiagonal eigenproblem, and provides facilities for achieving this accuracy for least squares problems. Future versions of LAPACK will also achieve this accuracy for other linear algebra problems, as discussed below.