Options
Ill-conditionedness need not be componentwise near to ill-posedness for least squares problems
Publikationstyp
Journal Article
Date Issued
1999-03
Sprache
English
Author(s)
Institut
TORE-URI
Journal
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.
Subjects
Componentwise distance
Condition number
Ill-posed
Least squares
Pseudoinverse
Underdetermined linear systems
DDC Class
004: Informatik
510: Mathematik