A Riemannian Newton Algorithm for Nonlinear Eigenvalue Problems

Zhi Zhao University of Macau Zheng-Jian Bai Xiamen University Xiao-Qing Jin University of Macau

Numerical Linear Algebra mathscidoc:1801.26002

SIAM Journal on Matrix Analysis, 36, (2), 752-774, 2015.6
We give the formulation of a Riemannian Newton algorithm for solving a class of nonlinear eigenvalue problems by minimizing a total energy function subject to the orthogonality constraint. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Moreover, the positive definiteness condition of the Riemannian Hessian of the total energy function at a solution is derived. Some numerical tests are reported to illustrate the efficiency of the proposed method for solving large-scale problems.
nonlinear eigenvalue problem, Riemannian Newton algorithm, Stiefel manifold, Grassmann manifold
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@inproceedings{zhi2015a,
  title={A Riemannian Newton Algorithm for Nonlinear Eigenvalue Problems},
  author={Zhi Zhao, Zheng-Jian Bai, and Xiao-Qing Jin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180103192246699427865},
  booktitle={SIAM Journal on Matrix Analysis},
  volume={36},
  number={2},
  pages={752-774},
  year={2015},
}
Zhi Zhao, Zheng-Jian Bai, and Xiao-Qing Jin. A Riemannian Newton Algorithm for Nonlinear Eigenvalue Problems. 2015. Vol. 36. In SIAM Journal on Matrix Analysis. pp.752-774. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180103192246699427865.
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