Unifying unitary and hyperbolic transformations

In this paper, we describe unified formulas for unitary and hyperbolic reflections and rotations, and show how these unified transformations can be used to compute a Hermitian triangular decomposition (R) over cap(H) D (R) over cap of a strongly nonsingular indefinite matrix (A) over cap given in the form (A) over cap = X-1(H) X-1 + alpha X-2(H) X-2, alpha = +/-1). The unification is achieved by the introduction of signature matrices which determine whether the applicable transformations are unitary, hyperbolic, or their generalizations. We derive formulas for the condition numbers of the unified transformations, propose pivoting strategics for lowering the condition number of the transformations, and present a unified stability analysis for applying the transformations to a matrix. (C) 2000 Elsevier Science Inc. All rights reserved.