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Volume 25, Issue 5
Fast Parallelizable Methods for Computing Invariant Subspaces of Hermitian Matrices

Zhenyue Zhang, Hongyuan Zha & Wenlong Ying

J. Comp. Math., 25 (2007), pp. 583-594.

Published online: 2007-10

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  • Abstract

We propose a quadratically convergent algorithm for computing the invariant subspaces of a Hermitian matrix. Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big $O$ notation. We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates. Several numerical examples are given which compare some aspects of the existing algorithms and the new algorithms.

  • Keywords

Eigenvalue, Invariant subspace, Hermitian matrix, QR method, Parallelizable method.

  • AMS Subject Headings

15A18, 65F05, 65F35.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-25-583, author = {Zhenyue Zhang, Hongyuan Zha and Wenlong Ying}, title = {Fast Parallelizable Methods for Computing Invariant Subspaces of Hermitian Matrices}, journal = {Journal of Computational Mathematics}, year = {2007}, volume = {25}, number = {5}, pages = {583--594}, abstract = {

We propose a quadratically convergent algorithm for computing the invariant subspaces of a Hermitian matrix. Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big $O$ notation. We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates. Several numerical examples are given which compare some aspects of the existing algorithms and the new algorithms.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8715.html} }
TY - JOUR T1 - Fast Parallelizable Methods for Computing Invariant Subspaces of Hermitian Matrices AU - Zhenyue Zhang, Hongyuan Zha & Wenlong Ying JO - Journal of Computational Mathematics VL - 5 SP - 583 EP - 594 PY - 2007 DA - 2007/10 SN - 25 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8715.html KW - Eigenvalue, Invariant subspace, Hermitian matrix, QR method, Parallelizable method. AB -

We propose a quadratically convergent algorithm for computing the invariant subspaces of a Hermitian matrix. Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big $O$ notation. We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates. Several numerical examples are given which compare some aspects of the existing algorithms and the new algorithms.

Zhenyue Zhang, Hongyuan Zha and Wenlong Ying. (2007). Fast Parallelizable Methods for Computing Invariant Subspaces of Hermitian Matrices. Journal of Computational Mathematics. 25 (5). 583-594. doi:
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