Purdue University Graduate School
vogel_on_the_structured_eigenvalue_problem.pdf (1.43 MB)

On the Structured Eigenvalue Problem: Methods, Analysis, and Applications

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posted on 2019-01-17, 14:04 authored by James P. VogelJames P. Vogel
This PhD thesis is an important development in the theories, methods, and applications of eigenvalue algorithms for structured matrices. Though eigenvalue problems have been well-studied, the class of matrices that admit very fast (near-linear time) algorithms was quite small until very recently. We developed and implemented a generalization of the famous symmetric tridiagonal divide-and-conquer algorithm to a much larger class of rank structured matrices (symmetric hierarchically semisperable, or HSS) that appear frequently in applications. Altogether, this thesis makes valuable contributions to three different major areas of scientific computing: algorithmic development, numerical analysis, and applications. In addition to the previously stated divide-and-conquer algorithm, we generalize to larger classes of eigenvalue problems and provide several key new low-rank update algorithms. A major contribution the analysis of the structured eigenvalue problem. In addition to standard perturbation analysis, we elucidate some subtle and previously under-examined issues in structured matrix eigenvalue problems such as subspace contributions and secular equation conditioning. Finally, several applications are studied.


Degree Type

  • Doctor of Philosophy


  • Mathematics

Campus location

  • West Lafayette

Advisor/Supervisor/Committee Chair

Jianlin Xia

Additional Committee Member 2

Venkataramanan Balakrishnan

Additional Committee Member 3

Jingwei Hu

Additional Committee Member 4

Peijun Li

Additional Committee Member 5

Jie Shen

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