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.
- Research Organization:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- W-7405-ENG-48
- OSTI ID:
- 1010828
- Report Number(s):
- LLNL-JRNL-426024; TRN: US201108%%532
- Journal Information:
- SIAM Journal on Matrix Analysis, vol. 31, no. 5, September 16, 2010, pp. 2650-2664, Vol. 31, Issue 5
- Country of Publication:
- United States
- Language:
- English
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