On Perturbations Of The Equality Constrained Stiffly Weighted Least Squares Problem

Variety of generalized least squares problems, such as the least squares problems, the total least squares problems, equality constrained least squares problems and stiffly weighted least squares problems, play important roles in computational mathematics. Many experts show great interests in this area. These problems have wide applications in statistic science, optimization, GPS, signal processing, computational physics, computational chemistry, geophysics and information science, etc.In this paper we focus on the equality constrained weighted least squares problems (WLSE). It is very important but difficult to study the supremum of the constrained weighted pseudo-inverses, the stability of the constrained weighted pseudo-inverses and the constrained weighted projections as well as to work out stable and high-precision numerical methods, which will be discussed in this paper. It is shown that a WLSE problem might be unstable even if the condition number of the coefficient matrix is small. When weighted matrices are given stiffly diagonal matrices, or from some real symmetric positive semidefinite matrices set, we obtain the perturbation bounds and stability conditions for the WLSE problems by studying the perturbations of the constrained weighted pseudo-inverses and the constrained weighted projections. Finally, we provide some numerical stable methods for computing WLSE problems.