Spectral Condition Numbers of Orthogonal Projections and Full Rank Linear Least Squares Residuals

Details Spectral Condition Numbers of Orthogonal Projections and Full Rank Linear Least Squares Residuals Spectral Condition Numbers of Orthogonal Projections and Full Rank Linear Least Squares Residuals

2010-03-13

arxiv.org/abs/1003.2737v5

SIAM J. Matrix Anal. Appl., vol. 31, no. 5, pp. 2934-949, 2010

Mathematics

Numerical Analysis

15 pages, 1 figure, 2 tables

Scientific paper

10.1137/090777773

A simple formula is proved to be a tight estimate for the condition number of the full rank linear least squares residual with respect to the matrix of least squares coefficients and scaled 2-norms. The tight estimate reveals that the condition number depends on three quantities, two of which can cause ill-conditioning. The numerical linear algebra literature presents several estimates of various instances of these condition numbers. All the prior values exceed the formula introduced here, sometimes by large factors.

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