The hyperbolic elimination method for solving the equality constrained indefinite least squares problem

摘要

Recently, Liu and Wang [Q. Liu and M. Wang, Algebraic properties and perturbation results for the indefinite least squares problem with equality constraints, Int. J. Comp. Math. 87(2) (2010), pp. 425-434.] proved that the solution of the equality constrained indefinite least squares (ILSE) problem min_(Bx=d)(b-Ax)~T J(b - Ax), J = diag(-I_q, I_p) is the limit of the solution of the unconstrained weighted indefinite least squares (WILS) problem min(f_μ-G_μx_μ)~T J (f_μ-G_μx_μ) with G_μ =(μb a), fμ=(μd b), and J = diag(I_s, J) as the weight μ tends to infinity, assuming that B has full row rank and x~T A~T JAx > 0 for all nonzero x ∈ null(B). Based on this observation, we derive a type of elimination method by applying the hyperbolic QR factorization method to above WILS problem and taking the limit analytically. Theoretical analysis shows that the method obtained is forward stable under a reasonable assumption. We illustrate our results with numerical tests.