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

Abstract

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 and [Jtilde]=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.

Keywords:

• equality constrained indefinite least squares problem
• hyperbolic QR factorization method
• hyperbolic elimination method
• forward stable

2000 AMS Subject Classifications:

• 65F20
• 65G05

Acknowledgements

This work was supported by Shanghai Leading Academic Discipline Project(J50101) and Shanghai Municipal Education Commission (06AZ088).