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Volume 3, Issue 3
On the Convergence of Diagonal Elements and Asymptotic Convergence Rates for the Shifted Tridiagonal QL Algorithm

Er-Xiong Jiang

J. Comp. Math., 3 (1985), pp. 252-261.

Published online: 1985-03

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  • Abstract

The convergence of diagonal elements of an irreducible symmetric triadiagonal matrix under QL algorithm with some kinds of shift is discussed. It is proved that if $\alpha_1-\sigma$→0 and $\beta_j$→0, j=1,2,...,m, then $\alpha_j$→$λ_j$ where $λ_j$ are m eigenvalues of the matrix, and $\sigma$ is the origin shift. The asymptotic convergence rates of three kinds of shift, Rayleigh quotient shift, Wilkinson's shift and RW shift, are analysed.  

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@Article{JCM-3-252, author = {Er-Xiong Jiang}, title = {On the Convergence of Diagonal Elements and Asymptotic Convergence Rates for the Shifted Tridiagonal QL Algorithm}, journal = {Journal of Computational Mathematics}, year = {1985}, volume = {3}, number = {3}, pages = {252--261}, abstract = {

The convergence of diagonal elements of an irreducible symmetric triadiagonal matrix under QL algorithm with some kinds of shift is discussed. It is proved that if $\alpha_1-\sigma$→0 and $\beta_j$→0, j=1,2,...,m, then $\alpha_j$→$λ_j$ where $λ_j$ are m eigenvalues of the matrix, and $\sigma$ is the origin shift. The asymptotic convergence rates of three kinds of shift, Rayleigh quotient shift, Wilkinson's shift and RW shift, are analysed.  

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9622.html} }
TY - JOUR T1 - On the Convergence of Diagonal Elements and Asymptotic Convergence Rates for the Shifted Tridiagonal QL Algorithm AU - Er-Xiong Jiang JO - Journal of Computational Mathematics VL - 3 SP - 252 EP - 261 PY - 1985 DA - 1985/03 SN - 3 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9622.html KW - AB -

The convergence of diagonal elements of an irreducible symmetric triadiagonal matrix under QL algorithm with some kinds of shift is discussed. It is proved that if $\alpha_1-\sigma$→0 and $\beta_j$→0, j=1,2,...,m, then $\alpha_j$→$λ_j$ where $λ_j$ are m eigenvalues of the matrix, and $\sigma$ is the origin shift. The asymptotic convergence rates of three kinds of shift, Rayleigh quotient shift, Wilkinson's shift and RW shift, are analysed.  

Er-Xiong Jiang. (1985). On the Convergence of Diagonal Elements and Asymptotic Convergence Rates for the Shifted Tridiagonal QL Algorithm. Journal of Computational Mathematics. 3 (3). 252-261. doi:
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