TABLE OF CONTENTS Most of LAPACK's condition numbers and error bounds are based on estimated condition numbers ... 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.
- E. ANDERSON et al., LAPACK Users' Guide (1999)
The importance of the counter-examples is that they make clear that any effort toward proving that the algorithms always produce useful estimations is fruitless. It may be possible to prove that the algorithms produce useful estimations in certain situations, however, and this should be pursued. An effort simply to construct more complex algorithms is dangerous.
- A. K. CLINE and R. K. REW, A Set of Counter-Examples to Three Condition Number Estimators (1983)
Singularity is almost invariably a clue.
- SIR ARTHUR CONAN DOYLE, The Boscombe Valley Mystery (1892)
Condition number estimation is the process of computing an inexpensive estimate of a condition number, where "inexpensive" usually means that the cost is an order of magnitude less than would be required to compute the condition number exactly. Condition estimates are required when evaluating forward error bounds for computed solutions to many types of linear algebra problems. An estimate of the condition number that is correct to within a factor 10 is usually acceptable, because it is the magnitude of an error bound that is of interest,...
