Abstract
Condition numbers of the full rank least squares (LS) problem minx||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 x0 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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