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<TITLE>Complete Orthogonal Decomposition for Weighted Least Squares</TITLE>
<H2>Complete Orthogonal Decomposition for Weighted Least Squares</H2>
 Patricia D. Hough and  Stephen A. Vavasis<BR>
95-203<BR>
December 1994
<p>
 Consider a full-rank weighted least squares problem in which the weight matrix is highly ill-conditioned. Because of the ill-conditioning, standard methods for solving least-squares problems, QR factorization and the nullspace method for example, break down. G.W. Stewart established a norm bound for such a system of equations, indicating that it may be possible to find an algorithm that gives an accurate solution. S.A. Vavasis proposed a new definition of stability that is based on this result. He also defined the NSH algorithm for solving this least-squares problem and showed that it satisfies his definition of stability. In this paper, we propose a complete orthogonal decomposition algorithm to solve this problem and show that it is also stable. This new algorithm is simpler and more efficient than the NSH method.<P>
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