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Numerical methods for eigenvalues of singular polynomial eigenvalue problems

Michiel E. Hochstenbach, Christian Mehl, Bor Plestenjak

Abstract

Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be generalized to treat singular polynomial eigenvalue problems. The common denominator of all three approaches is a transformation of a singular into a regular matrix polynomial whose eigenvalues are a disjoint union of the eigenvalues of the singular polynomial, called true eigenvalues, and additional fake eigenvalues. The true eigenvalues can then be separated from the fake eigenvalues using information on the corresponding left and right eigenvectors. We illustrate the approaches on several interesting applications, including bivariate polynomial systems and ZGV points.

Numerical methods for eigenvalues of singular polynomial eigenvalue problems

Abstract

Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be generalized to treat singular polynomial eigenvalue problems. The common denominator of all three approaches is a transformation of a singular into a regular matrix polynomial whose eigenvalues are a disjoint union of the eigenvalues of the singular polynomial, called true eigenvalues, and additional fake eigenvalues. The true eigenvalues can then be separated from the fake eigenvalues using information on the corresponding left and right eigenvectors. We illustrate the approaches on several interesting applications, including bivariate polynomial systems and ZGV points.

Paper Structure

This paper contains 15 sections, 10 theorems, 53 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Three methods
  5. Comparison to linearization
  6. Applications and numerical examples
  7. Polynomial equations
  8. ZGV points
  9. Set of singular QEP examples
  10. Other applications
  11. Incorrectly determined rank
  12. Rectangular matrix polynomials
  13. Computational complexity
  14. Conclusions
  15. Simple random eigenvalues

Key Result

Theorem 2.1

Let $P(\lambda)$ be an $m\times n$ matrix polynomial of normal rank $r$. Then there exist unimodular matrix polynomials $E(\lambda)$ and $F(\lambda)$ of sizes $m\times m$ and $n\times n$, respectively, such that where $d_1(\lambda), \dotsc, d_r(\lambda)$ are monic polynomials such that $d_j(\lambda)$ is a divisor of $d_{j+1}(\lambda)$ for $j=1, \dotsc,r-1$. The $m\times n$ diagonal matrix polynom

Figures (1)

  • Figure 1: Real dispersion curves $\omega(k)$ and ZGV points of Example \ref{['ex:simple_QEP_ZGV']}

Theorems & Definitions (32)

  • Theorem 2.1: Smith form
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Index sum theorem
  • Theorem 2.7
  • Definition 2.8
  • Definition 2.9
  • Lemma 3.1
  • ...and 22 more