Direction-preserving and Schur-monotonic Semi-separable Approximations of Symmetric Positive Definite Matrices

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For a given symmetric positive definite matrix A {element_of} R{sup nxn}, we develop a fast and backward stable algorithm to approximate A by a symmetric positive-definite semi-separable matrix, accurate to any prescribed tolerance. In addition, this algorithm preserves the product, AZ, for a given matrix Z {element_of} R{sup nxd}, 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 ... continued below

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Gu, Ming; Li, Xiaoye Sherry & Vassilevski, Panayot S. October 20, 2009.

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For a given symmetric positive definite matrix A {element_of} R{sup nxn}, we develop a fast and backward stable algorithm to approximate A by a symmetric positive-definite semi-separable matrix, accurate to any prescribed tolerance. In addition, this algorithm preserves the product, AZ, for a given matrix Z {element_of} R{sup nxd}, 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.

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  • Journal Name: SIAM Journal ofMatrix Analysis and Applications

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  • Report No.: LBNL-2771E
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 974541
  • Archival Resource Key: ark:/67531/metadc929360

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  • October 20, 2009

Added to The UNT Digital Library

  • Nov. 13, 2016, 7:26 p.m.

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  • Nov. 18, 2016, 2:35 p.m.

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Gu, Ming; Li, Xiaoye Sherry & Vassilevski, Panayot S. Direction-preserving and Schur-monotonic Semi-separable Approximations of Symmetric Positive Definite Matrices, article, October 20, 2009; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc929360/: accessed December 10, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.