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The shift-and-invert Arnoldi method for singular matrix pencils

Karl Meerbergen, Zhijun Wang

Abstract

The numerical solution of singular generalized eigenvalue problems is still challenging. In Hochstenbach, Mehl, and Plestenjak, Solving Singular Generalized Eigenvalue Problems by a Rank-Completing Perturbation, SIMAX 2019, a rank-completing perturbation was proposed and a related bordering of the singular pencil. For large sparse pencils, we propose an LU factorization that determines a rank completing perturbation that regularizes the pencil and that is then used in the shift-and-invert Arnoldi method to obtain eigenvalues nearest a shift. Numerical examples illustrate the theory and the algorithms.

The shift-and-invert Arnoldi method for singular matrix pencils

Abstract

The numerical solution of singular generalized eigenvalue problems is still challenging. In Hochstenbach, Mehl, and Plestenjak, Solving Singular Generalized Eigenvalue Problems by a Rank-Completing Perturbation, SIMAX 2019, a rank-completing perturbation was proposed and a related bordering of the singular pencil. For large sparse pencils, we propose an LU factorization that determines a rank completing perturbation that regularizes the pencil and that is then used in the shift-and-invert Arnoldi method to obtain eigenvalues nearest a shift. Numerical examples illustrate the theory and the algorithms.

Paper Structure

This paper contains 15 sections, 6 theorems, 53 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kronecker canonical form
  4. Bordered system and spectral properties
  5. Arnoldi method
  6. Arnoldi method for bordered systems
  7. Two-sided Arnoldi
  8. LU factorization of singular matrix and rank detection
  9. Rectangular pencils
  10. Numerical examples
  11. Tolerance for the LU factorization
  12. Multiparameter eigenvalue problems
  13. Quadratic eigenvalue problem
  14. Rectangular eigenvalue problems
  15. Conclusions

Key Result

Proposition 2.2

\newlabelborder0 For singular matrix pencil $A-\lambda B$ of normal rank $k$, eq:BGEP with the conditions $U^*{\bf x}=0$ and $V^*{\bf y}=0$, where $U,V\in\mathbb{C}^{n\times n-k}$ are random matrices, yields regular eigenvalues of pencil $A-\lambda B$.

Figures (2)

  • Figure 1: 13-degree-of-freedom Truss
  • Figure 2: elements of 13-degree-of-freedom Truss

Theorems & Definitions (15)

  • Example 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proof 1
  • Example 2.4
  • Definition 4.1: Permutation matrix
  • Lemma 4.2
  • Proof 2
  • Lemma 4.3
  • Proof 3
  • ...and 5 more