Iterative refinement of linear least squares solutions II

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