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Fast structured LU factorization for nonsymmetric matrices

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Abstract

In this paper, an approximate LU factorization algorithm is developed for nonsymmetric matrices based on the hierarchically semiseparable matrix techniques. It utilizes a technique involving orthogonal transformations and approximations to avoid the explicit computation of the Schur complement in each factorization step. A modified compression method is further developed for the case when some diagonal blocks have small singular values. The complexity of the methods proposed in this paper is analyzed and shown to be \(O(N^2k)\), where \(N\) is the dimension of matrix and \(k\) is the maximum off-diagonal (numerical) rank. Depending on the accuracy and efficiency requirements in the approximation, this factorization can be used either as a direct solver or a preconditioner. Numerical results from applications are included to show the efficiency of our method.

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Acknowledgments

The authors are very grateful to the referees for their valuable suggestions.

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Authors and Affiliations

  1. College of Science, National University of Defense Technology, Changsha, 410073, China

    Shengguo Li & Lizhi Cheng

  2. Department of Mathematics, University of California, Berkeley, CA, 47920, USA

    Ming Gu

Authors
  1. Shengguo Li
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  2. Ming Gu
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  3. Lizhi Cheng
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Corresponding author

Correspondence to Shengguo Li.

Additional information

This paper is dedicated to Professor Lothar Reichel on the occasion of his 60th birthday.

The work of Gu was supported in part by NSF Award CCF-0830764 and by the DOE Office of Advanced Scientific Computing Research under contact number DE-AC02-05CH11231. The work of Cheng was supported by Natural Science Project of NUDT (No. JC120201) and NSF of Hunan Province in China (No. 13JJ2001).

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Li, S., Gu, M. & Cheng, L. Fast structured LU factorization for nonsymmetric matrices. Numer. Math. 127, 35–55 (2014). https://doi.org/10.1007/s00211-013-0582-0

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  • DOI: https://doi.org/10.1007/s00211-013-0582-0

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