Abstract
Condition numbers of the full rank least squares (LS) problem min x ||Ax − b||2 are considered theoretically and their computational implementation is compared. These condition numbers range from a simple normwise measure that may overestimate by several orders of magnitude the true numerical condition of the LS problem, to refined componentwise and normwise measures. Inequalities that relate these condition numbers are established, and it is concluded that the solution x 0 of the LS problem may be well-conditioned in the normwise sense, even if one of its components is ill-conditioned. It is shown that the refined condition numbers are ill-conditioned in some circumstances, the cause of this ill-conditioning is identified, and its implications are discussed.
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Winkler, J.R. (2005). A Comparison of Condition Numbers for the Full Rank Least Squares Problem. In: Winkler, J., Niranjan, M., Lawrence, N. (eds) Deterministic and Statistical Methods in Machine Learning. DSMML 2004. Lecture Notes in Computer Science(), vol 3635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11559887_18
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DOI: https://doi.org/10.1007/11559887_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29073-5
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