Abstract
The G-algorithm was proposed by Bareiss [1] as a method for solving the weighted linear least squares problem. It is a square root free algorithm similar to the fast Givens method except that it triangularizes a rectangular matrix a column at a time instead of one element at a time.
In this paper an error analysis of the G-algorithm is presented which shows that it is as stable as any of the standard orthogonal decomposition methods for solving least squares problems. The algorithm is shown to be a competitive method for sparse least squares problems.
A pivoting strategy is given for heavily weighted problems similar to that in [14] for the Householder-Golub algorithm. The strategy is prohibitively expensive, but it is not necessary for most of the least squares problems that arise in practice.
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The research was supported by the National Science Foundation under contract no. MCS-8201065 and by the Office of Naval Research under contract no. N0014-80-0517.
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Barlow, J.L. Stability analysis of the G-algorithm and a note on its application to sparse least squares problems. BIT 25, 507–520 (1985). https://doi.org/10.1007/BF01935371
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DOI: https://doi.org/10.1007/BF01935371