An algorithm and a stability theory for downdating the ULV decomposition

Abstract

An alternative to performing the singular value decomposition is to factor a matrixA into $$A = U\left( {\begin{array}{*{20}c} C \\ 0 \\ \end{array} } \right)V^T $$ , whereU andV are orthogonal matrices andC is a lower triangular matrix which indicates a separation between two subspaces by the size of its columns. These subspaces are denoted byV = (V 1,V 2), where the columns ofC are partitioned conformally intoC = (C 1,C 2) with ‖C 2 ‖ F ≤ e. Here e is some tolerance. In recent years, this has been called the ULV decomposition (ULVD). If the matrixA results from statistical observations, it is often desired to remove old observations, thus deleting a row fromA and its ULVD. In matrix terms, this is called a downdate. A downdating algorithm is proposed that preserves the structure in the downdated matrix $$\bar C$$ to the extent possible. Strong stability results are proven for these algorithms based upon a new perturbation theory.