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

Authors:
Ming Gu;Xiaoye S. Li;Panayot S. Vassilevski
Affiliations:
mgu@math.berkeley.edu;xsli@lbl.gov;panayot@llnl.gov
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
2010

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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.