Complete Orthogonal Decomposition for Weighted Least Squares

This paper proposes a complete orthogonal decomposition (COD) algorithm for solving weighted least-squares problems. In applications, the weight matrix can be highly ill conditioned, and this can cause standard methods like QR factorization to return inaccurate answers in floating-point arithmetic. Stewart and Todd independently established a norm bound for the weighted least-squares problem that is independent of the weight matrix. Vavasis proposed a definition of a "stable" solution of weighted least squares based on this norm bound: The solution computed by a stable algorithm must satisfy an accuracy bound that is not affected by ill conditioning in the weight matrix. A forward error analysis shows that the COD algorithm is stable in this sense, but it is simpler and more efficient than the algorithm proposed by Vavasis. Our forward error bound is contrasted to the backward error analysis of other previous works on weighted least squares.