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A positivity preserving inexact Noda iteration for computing the smallest eigenpair of a large irreducible \(M\)-matrix

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Abstract

In this paper, based on the Noda iteration, we present inexact Noda iterations (INI), to find the smallest eigenvalue and the associated positive eigenvector of a large irreducible nonsingular \(M\)-matrix. The positivity of approximations is critical in applications, and if the approximations lose the positivity then they may be meaningless and could not be interpreted. We propose two different inner tolerance strategies for solving the inner linear systems involved, and prove that the convergence of resulting INI algorithms is globally linear and superlinear with the convergence order \(\frac{1+\sqrt{5}}{2}\), respectively. The proposed INI algorithms are structure preserving and maintains the positivity of approximate eigenvectors. We also revisit the exact Noda iteration and establish a new quadratic convergence result. All the above is first done for the problem of computing the Perron root and the positive Perron vector of an irreducible nonnegative matrix and is then adapted to computing the smallest eigenpair of the irreducible nonsingular \(M\)-matrix. Numerical examples illustrate that the proposed INI algorithms are practical, and they always preserve the positivity of approximate eigenvectors. We compare them with the Jacobi–Davidson method, the implicitly restarted Arnoldi method and the explicitly restarted Krylov–Schur method, all of which cannot guarantee the positivity of approximate eigenvectors, and illustrate that the overall efficiency of the INI algorithms is competitive with and can be considerably higher than the latter three methods.

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Notes

  1. We thank one of the referees, who suggested to us this more direct proof than our original one.

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Acknowledgments

We thank the referees for their careful reading of our paper and a number of valuable comments, which helped us improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Zhongxiao Jia

  2. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan

    Wen-Wei Lin & Ching-Sung Liu

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  1. Zhongxiao Jia
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  2. Wen-Wei Lin
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  3. Ching-Sung Liu
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Correspondence to Zhongxiao Jia.

Additional information

Z. Jia was supported in part by the National Basic Research Program of China 2011CB302400 and the National Science Foundation of China (No. 11371219), and W.-W. Lin and C.-S. Liu were supported in part by the National Science Council, the National Center for Theoretical Sciences, the Center of MMSC, and ST Yau Center at Chiao-Da in Taiwan.

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Jia, Z., Lin, WW. & Liu, CS. A positivity preserving inexact Noda iteration for computing the smallest eigenpair of a large irreducible \(M\)-matrix. Numer. Math. 130, 645–679 (2015). https://doi.org/10.1007/s00211-014-0677-2

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  • DOI: https://doi.org/10.1007/s00211-014-0677-2

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