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A Riemannian under-determined BFGS method for least squares inverse eigenvalue problems

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Abstract

This paper is concerned with the parameterized least squares inverse eigenvalue problems for the case that the number of parameters to be constructed is greater than the number of prescribed realizable eigenvalues. Intrinsically, this is a specific problem of finding a zero of an under-determined nonlinear map defined between a Riemannian product manifold and a matrix space. To solve this problem, we propose a Riemannian under-determined BFGS algorithm with a specialized update formula for iterative linear operators, and an Armijo type line search is used. Global convergence properties of this algorithm are established under some mild assumptions. In addition, we also generalize a Riemannian inexact Newton method for solving this problem. Specially, the explicit form of the inverse of the linear operator corresponding to the perturbed normal Riemannian Newton equation is obtained, which improves the efficiency of Riemannian inexact Newton method. Numerical experiments are provided to illustrate the efficiency of the proposed method.

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Acknowledgements

We are very grateful to the two anonymous referees for their valuable comments on the paper, which have considerably improved the paper.

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Authors and Affiliations

  1. Department of Mathematics, School of Sciences, Hangzhou Dianzi University, Hangzhou, 310018, People’s Republic of China

    Zhi Zhao

  2. Department of Mathematics, University of Macau, Macao, People’s Republic of China

    Xiao-Qing Jin

  3. Department of Mathematics, School of Sciences, Zhejiang University of Science and Technology, Hangzhou, 310023, People’s Republic of China

    Teng-Teng Yao

Authors
  1. Zhi Zhao
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  2. Xiao-Qing Jin
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  3. Teng-Teng Yao
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Correspondence to Teng-Teng Yao.

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Communicated by Daniel Kressner.

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This work was supported by NSFC Nos. 11601112, 11671337, 11701514. The research of Z. Zhao was partially supported by the Zhejiang Provincial Natural Science Foundation of China (No. LY21A010010) and the Fundamental Research Funds for the Provincial Universities of Zhejiang (No. GK199900299012-006). The research of X. Q. Jin was supported by the research grants MYRG2019-00042-FST from University of Macau and 0014/2019/A from FDCT of Macao. The research of T. T. Yao was supported by the Zhejiang Provincial Natural Science Foundation of China (No. LY21A010004).

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Zhao, Z., Jin, XQ. & Yao, TT. A Riemannian under-determined BFGS method for least squares inverse eigenvalue problems. Bit Numer Math 62, 311–337 (2022). https://doi.org/10.1007/s10543-021-00874-z

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  • DOI: https://doi.org/10.1007/s10543-021-00874-z

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