DIRECTION-PRESERVING AND SCHUR-MONOTONIC SEMISEPARABLE APPROXIMATIONS OF SYMMETRIC POSITIVE DEFINITE MATRICES

作者：Gu Ming^*; Li Xiaoye S; Vassilevski Panayot S

来源：SIAM Journal on Matrix Analysis and Applications, 2010, 31(5): 2650-2664.

DOI：10.1137/090774331

• semiseparable matrix factorization
• positivity-preserving
• direction-preserving

For a given symmetric positive definite matrix A is an element of R(NxN), 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 is an element of R(Nxd), where d << N. Our algorithm guarantees the positivedefiniteness 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.

• 出版日期2010