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ARNOLDI TYPE ALGORITHMS FOR LARGE UNSYMMETRICmultiple eigenvalue PROBLEMS

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Journal of Computational Mathematics 1999年第3期17卷 257-274页

作者： Zhong-xiao Jia(Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China) 大连理工大学应用数学系辽宁大连

As is well known, solving matrix multiple eigenvalue problems is a very difficult topic. In this paper, Arnoldi type algorithms are proposed for large unsymmetric multiple eigenvalue problems when the matrix A involved is diagonalizable. The theoretical background is established, in which lower and upper error bounds for eigenvectors are new for both Arnoldi's method and a general perturbation problem, and furthermore these bounds are shown to be optimal and they generalize a classical perturbation bound due to W. Kahan in 1967 for A symmetric. The algorithms can adaptively determine the multiplicity of an eigenvalue and a basis of the associated eigenspace. Numerical experiments show reliability of the algorithms.

关键词： Arnoldi's process large unsymmetric matrix multiple eigenvalue diagonalizable error bounds

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Sensitivity Analysis of Semi-simple eigenvalues of Regular Quadratic eigenvalue Problems

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Acta Mathematicae Applicatae Sinica 2015年第2期31卷 499-518页

作者： Hui-qing XIE Department of Mathematics East China University of Science and Technology

This paper discusses the sensitivity analysis of semisimple eigenvalues and associated eigen-matrix triples of regular quadratic eigenvalue problems analytically dependent on several parameters. The directional derivatives of semisimple eigenvalues are obtained. The average of semisimple eigenvalues and corresponding eigen-matrix triple are proved to be analytic, and their partial derivatives are given. On these grounds, the sensitivities of the semisimple eigenvalues and corresponding eigenvector matrices are defined.

关键词： sensitivity analysis quadratic eigenvalue problems multiple eigenvalue matrix-valued functions

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IMPLICIT DETERMINANT METHOD FOR SOLVING AN HERMITIAN eigenvalue OPTIMIZATION PROBLEM

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Journal of Computational Mathematics 2023年第6期41卷 1117-1136页

作者： Siru Gong Yangfeng Su School of Mathematical Sciences Fudan UniversityShanghai 200433China

Implicit determinant method is an effective method for some linear eigenvalue optimization problems since it solves linear systems of equations rather than *** this paper,we generalize the implicit determinant method to solve an Hermitian eigenvalue optimization problem for smooth case and non-smooth *** prove that the implicit determinant method converges locally and *** experiments confirm our theoretical results and illustrate the efficiency of implicit determinant method.

关键词： eigenvalue optimization multiple eigenvalue Non-smooth optimization Implicit determinant method Crawford number

ON SMOOTH LU DECOMPOSITIONS WITH APPLICATIONS TO SOLUTIONS OF NONLINEAR eigenvalue PROBLEMS

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Journal of Computational Mathematics 2010年第6期28卷 745-766页

作者： Hua Dai Zhong-Zhi Bai Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing 210016 China LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

We study the smooth LU decomposition of a given analytic functional A-matrix A（A） and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A（A）, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.

关键词： Matrix-valued function Smooth LU decomposition Pivoting Nonlinear eigenvalue problem multiple eigenvalue Newton method.

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