An algorithm and a stability theory for downdating the ULV decomposition

An alternative to performing the singular value decomposition is to factor a matrix A into matrix presented, where U and V are orthogonal matrices and C is a lower triangular matrix which indicates a separation between two subspaces by the size of its columns. These subspaces are denoted by V = (V₁, V₂), where the columns of C are partitioned conformally into C = (C₁, C₂) with ∥ C₂ ∥F≤ ε . Here ε is some tolerance. In recent years, this has been called the ULV decomposition (ULVD). If the matrix A results from statistical observations, it is often desired to remove old observations, thus deleting a row from A and its ULVD. In matrix terms, this is called a downdate. A downdating algorithm is proposed that preserves the structure in the downdated matrix C̄ to the extent possible. Strong stability results are proven for these algorithms based upon a new perturbation theory.