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Improved Error Bounds

 

The standard error analysis just outlined has a drawback: by using the infinity norm tex2html_wrap_inline18154 to measure the backward error, entries of equal magnitude in tex2html_wrap_inline18009 contribute equally to the final error bound tex2html_wrap_inline18158. This means that if z is sparse or has some tiny entries, a normwise backward stable algorithm may make large changes in these entries compared wit 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: tex2html_wrap_inline18164. The usual error bound is approximately tex2html_wrap_inline18166, which can arbitrarily overestimate the true error, tex2html_wrap_inline17202, if at least one tex2html_wrap_inline18170 is tiny and another one is large.

LAPACK addresses this inadequacy by providing some algorithms whose backward error tex2html_wrap_inline18009 is a tiny relative change in each component of z: tex2html_wrap_inline18176. 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.gif          

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 [4].

When solving Ax=b using expert driver PxyySVX or computational routine PxyyRFS, for example, we almost always compute tex2html_wrap_inline17482 satisfying tex2html_wrap_inline18075, where tex2html_wrap_inline18186 is a small relative change in tex2html_wrap_inline18188 and tex2html_wrap_inline18190 is a small relative change in tex2html_wrap_inline18192. In particular, if A is diagonal, the corresponding error bound is always tiny, as one would expect (see the next section).

ScaLAPACK 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.


next up previous contents index
Next: Error Bounds for Linear Up: Further Details: How Error Previous: Standard Error Analysis

Susan Blackford
Tue May 13 09:21:01 EDT 1997