Volume 11, Issue 4
Computing the Maximal Eigenpairs of Large Size Tridiagonal Matrices with $\mathcal{O}(1)$ Number of Iterations

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 877-894.

Published online: 2018-06

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• Abstract

In a series of papers, Chen [4–6] developed some efficient algorithms for computing the maximal eigenpairs for tridiagonal matrices. The key idea is to explicitly construct effective initials for the maximal eigenpairs and also to employ a self-closed iterative algorithm. In this paper, we extend Chen's algorithm to deal with large scale tridiagonal matrices with super-/sub-diagonal elements. By using appropriate scalings and by optimizing numerical complexity, we make the computational cost for each iteration to be $\mathcal{O}$($N$). Moreover, to obtain accurate approximations for the maximal eigenpairs, the total number of iterations is found to be independent of the matrix size, i.e., $\mathcal{O}$($1$) number of iterations. Consequently, the total cost for computing the maximal eigenpairs is $\mathcal{O}$($N$). The effectiveness of the proposed algorithm is demonstrated by numerical experiments.

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@Article{NMTMA-11-877, author = {}, title = {Computing the Maximal Eigenpairs of Large Size Tridiagonal Matrices with $\mathcal{O}(1)$ Number of Iterations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {11}, number = {4}, pages = {877--894}, abstract = {

In a series of papers, Chen [4–6] developed some efficient algorithms for computing the maximal eigenpairs for tridiagonal matrices. The key idea is to explicitly construct effective initials for the maximal eigenpairs and also to employ a self-closed iterative algorithm. In this paper, we extend Chen's algorithm to deal with large scale tridiagonal matrices with super-/sub-diagonal elements. By using appropriate scalings and by optimizing numerical complexity, we make the computational cost for each iteration to be $\mathcal{O}$($N$). Moreover, to obtain accurate approximations for the maximal eigenpairs, the total number of iterations is found to be independent of the matrix size, i.e., $\mathcal{O}$($1$) number of iterations. Consequently, the total cost for computing the maximal eigenpairs is $\mathcal{O}$($N$). The effectiveness of the proposed algorithm is demonstrated by numerical experiments.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2018.s11}, url = {http://global-sci.org/intro/article_detail/nmtma/12477.html} }
TY - JOUR T1 - Computing the Maximal Eigenpairs of Large Size Tridiagonal Matrices with $\mathcal{O}(1)$ Number of Iterations JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 877 EP - 894 PY - 2018 DA - 2018/06 SN - 11 DO - http://doi.org/10.4208/nmtma.2018.s11 UR - https://global-sci.org/intro/article_detail/nmtma/12477.html KW - AB -

In a series of papers, Chen [4–6] developed some efficient algorithms for computing the maximal eigenpairs for tridiagonal matrices. The key idea is to explicitly construct effective initials for the maximal eigenpairs and also to employ a self-closed iterative algorithm. In this paper, we extend Chen's algorithm to deal with large scale tridiagonal matrices with super-/sub-diagonal elements. By using appropriate scalings and by optimizing numerical complexity, we make the computational cost for each iteration to be $\mathcal{O}$($N$). Moreover, to obtain accurate approximations for the maximal eigenpairs, the total number of iterations is found to be independent of the matrix size, i.e., $\mathcal{O}$($1$) number of iterations. Consequently, the total cost for computing the maximal eigenpairs is $\mathcal{O}$($N$). The effectiveness of the proposed algorithm is demonstrated by numerical experiments.

Tao Tang & Jiang Yang. (2020). Computing the Maximal Eigenpairs of Large Size Tridiagonal Matrices with $\mathcal{O}(1)$ Number of Iterations. Numerical Mathematics: Theory, Methods and Applications. 11 (4). 877-894. doi:10.4208/nmtma.2018.s11
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