Direction-Preserving and Schur-Monotonic Semi-Separable Approximations of Symmetric Positive Definite Matrices}

For a given symmetric positive definite matrix A {element_of} R{sup N x N}, we develop a fast and backward stable algorithm to approximate A by a symmetric positive-definite semi-separable matrix, accurate to a constant multiple of any prescribed tolerance. In addition, this algorithm preserves the product, AZ, for a given matrix Z {element_of} R{sup N x d}, where d << N. Our algorithm guarantees the positive-definiteness of the semi-separable matrix by embedding an approximation strategy inside a Cholesky factorization procedure to ensure that the Schur complements during the Cholesky factorization all remain positive definite after approximation. It uses a robust direction-preserving approximation scheme to ensure the preservation of AZ. We present numerical experiments and discuss potential implications of our work