Iterative refinement of linear least squares solutions I

An iterative procedure is developed for reducing the rounding errors in the computed least squares solution to an overdetermined system of equationsAx =b, whereA is anm ×n matrix (m ≧n) of rankn. The method relies on computing accurate residuals to a certain augmented system of linear equations, by using double precision accumulation of inner products. To determine the corrections, two methods are given, based on a matrix decomposition ofA obtained either by orthogonal Householder transformations or by a modified Gram-Schmidt orthogonalization. It is shown that the rate of convergence in the iteration is independent of the right hand side,b, and depends linearly on the condition number, ℳ2135;(A), of the rectangular matrixA. The limiting accuracy achieved will be approximately the same as that obtained by a double precision factorization.In a second part of this paper the case whenx is subject to linear constraints and/orA has rank less thann is covered. Here also ALGOL-programs embodying the derived algorithms will be given.