Solving the indefinite least squares problem by hyperbolic QR factorization

The indefinite least squares (ILS) problem involves minimizing a certain type of indefinite quadratic form. We develop perturbation theory for the problem and identify a condition number. We describe and analyze a method for solving the ILS problem based on hyperbolic QR factorization. This method has a lower operation count than one recently proposed by Chandrasekaran, Gu, and Sayed that employs both QR and Cholesky factorizations. We give a rounding error analysis of the new method and use the perturbation theory to show that under a reasonable assumption the method is forward stable. Our analysis is quite general and sheds some light on the stability properties of hyperbolic transformations. In our numerical experiments the new method is just as accurate as the method of Chandrasekaran, Gu, and Sayed.