Unifying Unitary and Hyperbolic Rotations and Reflectors

In Cholesky updating, Givens rotations or Householder transformations are used. In Cholesky downdating, where the matrix resulted from downdating is positive definite, hyperbolic rotations or hyperbolic Householder transformations are used. There are applications, for example, fast Toeplitz triangularization, where the matrix resulted from downdating may be indefinite. In this paper, we unify the unitary and hyperbolic rotations and Householder transformations (reflectors) and present a unified algorithm for both updating and general downdating where the downdated matrix can be either positive definite or indefinite. Key Words. Cholesky downdating, hyperbolic rotation, hyperbolic Householder transformation, fast Toeplitz triangularization. 1 Introduction Consider an n-by-n Hermitian and positive definite matrix A and its Cholesky factorization A = R H R (1) where R is upper triangular. The Cholesky downdating problem is to compute the Cholesky factorization of a new matrix A obtai..