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

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

Published online: 2007-10

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• Abstract

We propose a {\it quadratically} convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of {\it one} matrix-matrix multiplication and {\it 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

15A18 65F05 65F35.

@Article{JCM-25-583, author = {}, 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 {\it quadratically} convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of {\it one} matrix-matrix multiplication and {\it 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 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 KW - Invariant subspace KW - Hermitian matrix KW - QR method KW - Parallelizable method AB - We propose a {\it quadratically} convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of {\it one} matrix-matrix multiplication and {\it 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.