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# Ill-conditionedness need not be componentwise near to ill-posedness for least squares problems

Publikationstyp

Journal Article

Publikationsdatum

1999-03

Sprache

English

Author

Institut

TORE-URI

Enthalten in

Volume

39

Issue

1

Start Page

143

End Page

151

Citation

BIT Numerical Mathematics 39 (1): 143-151 (1999-03)

Publisher DOI

Scopus ID

Publisher

Swets & Zeitlinger

The condition number of a problem measures the sensitivity of the answer to small changes in the input, where "small" refers to some distance measure. A problem is called ill-conditioned if the condition number is large, and it is called ill-posed if the condition number is infinity. It is known that for many problems the (normwise) distance to the nearest ill-posed problem is proportional to the reciprocal of the condition number. Recently it has been shown that for linear systems and matrix inversion this is also true for componentwise distances. In this note we show that this is no longer true for least squares problems and other problems involving rectangular matrices. Problems are identified which are arbitrarily ill-conditioned (in a componentwise sense) whereas any componentwise relative perturbation less than 1 cannot produce an ill-posed problem. Bounds are given using additional information on the matrix.