Conditioning in the Application of -orthogonal Transformations

This work attempts to give a uniied treatment of sensitivity issues in problems involving the application of-orthogonal transformations. Algorithms for such problems are sometimes implemented with minor variations in the way elementary hyperbolic transformations are used to construct a-orthogonal transformation. This results in a possible variation in the condition of the elementary transformations which can obscure the condition of the underlying problem. To characterize the possible elementary transformations which can be used to solve a given problem and to clarify important conditioning issues, we introduce a canonical decomposition of a partitioned-orthogonal matrix which is analogous to the CS decomposition of a partitioned orthogonal matrix. We then proceed to prove optimality properties of the hyperbolic transformations given by the decomposition and show how these properties relate to the sensitivity of Cholesky downdating and block Toeplitz factorization problems.