Direction-Preserving and Schur-Monotonic Semiseparable Approximations of Symmetric Positive Definite Matrices

Abstract

For a given symmetric positive definite matrix $A\in\mathbf{R}^{N\times N}$, we develop a fast and backward stable algorithm to approximate A by a symmetric positive definite semiseparable matrix, accurate to a constant multiple of any prescribed tolerance. In addition, this algorithm preserves the product, $AZ$, for a given matrix $Z\in\mathbf{R}^{N\times d}$, where $d\ll N$. Our algorithm guarantees the positive-definiteness of the semiseparable 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 the potential implications of our work.