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Riemannian inexact Newton method for structured inverse eigenvalue and singular value problems

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Abstract

Inverse eigenvalue and singular value problems have been widely discussed for decades. The well-known result is the Weyl-Horn condition, which presents the relations between the eigenvalues and singular values of an arbitrary matrix. This result by Weyl-Horn then leads to an interesting inverse problem, i.e., how to construct a matrix with desired eigenvalues and singular values. In this work, we do that and more. We propose an eclectic mix of techniques from differential geometry and the inexact Newton method for solving inverse eigenvalue and singular value problems as well as additional desired characteristics such as nonnegative entries, prescribed diagonal entries, and even predetermined entries. We show theoretically that our method converges globally and quadratically, and we provide numerical examples to demonstrate the robustness and accuracy of our proposed method.

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Acknowledgements

This research work is also supported by the National Center for Theoretical Sciences in Taiwan. The authors wish to thank Prof. Michiel E. Hochstenbach for his highly valuable comments. They also thank Prof. Zheng-Jian Bai and Dr. Zhi Zhao for helpful discussions.

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Authors and Affiliations

  1. Center for General Education, National Formosa University, Huwei, 632, Taiwan

    Chun-Yueh Chiang

  2. Department of Mathematics, National Cheng Kung University, Tainan, 701, Taiwan

    Matthew M. Lin

  3. Department of Mathematics, University of Macau, Macao, China

    Xiao-Qing Jin

Authors
  1. Chun-Yueh Chiang
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  2. Matthew M. Lin
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  3. Xiao-Qing Jin
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Correspondence to Matthew M. Lin.

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Communicated by Michiel E. Hochstenbach.

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Chun-Yueh Chiang: This research was supported in part by the Ministry of Science and Technology of Taiwan under Grant 107-2115-M-150-002. Matthew M. Lin: This research was supported in part by the Ministry of Science and Technology of Taiwan under Grants 107-2115-M-006-007-MY2 and 108-2636-M-006-006. Xiao-Qing Jin: This research was supported in part by the research Grant MYRG2016-00077-FST from University of Macau.

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Chiang, CY., Lin, M.M. & Jin, XQ. Riemannian inexact Newton method for structured inverse eigenvalue and singular value problems. Bit Numer Math 59, 675–694 (2019). https://doi.org/10.1007/s10543-019-00754-7

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