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A Hidden Variable Resultant Method for the Polynomial Multiparameter Eigenvalue Problem

Emil Graf, Alex Townsend

TL;DR

The paper tackles solving polynomial multiparameter eigenvalue problems (PMEPs) by introducing a hidden variable tensor Dixon resultant framework that reduces PMEPs to univariate polynomial eigenvalue problems (PEPs) solvable by standard linearizations. It develops a complete generic-PME solver that leverages hidden-variable resultants and operator-determinant ideas to extract all coordinates of PMEP solutions from PEP eigenpairs, with residual checks to filter spurious results. Theoretical results establish conditions under which the tensor Dixon resultant correctly encodes PMEP solutions for generic systems, while practical strategies address nongeneric cases, including singular pencils and repeated eigenvalues. Numerical experiments in aeroelastic flutter and leaky-wave contexts demonstrate high accuracy and robustness, illustrating the method's broad applicability and potential to replace bespoke linearizations in many PMEP applications.

Abstract

We present a novel, global algorithm for solving polynomial multiparameter eigenvalue problems (PMEPs) by leveraging a hidden variable tensor Dixon resultant framework. Our method transforms a PMEP into one or more univariate polynomial eigenvalue problems, which are solved as generalized eigenvalue problems. Our general approach avoids the need for custom linearizations of PMEPs. We provide rigorous theoretical guarantees for generic PMEPs and give practical strategies for nongeneric systems. Benchmarking on applications from aeroelastic flutter and leaky wave propagation confirms that our algorithm attains high accuracy and robustness while being broadly applicable to many PMEPs.

A Hidden Variable Resultant Method for the Polynomial Multiparameter Eigenvalue Problem

TL;DR

The paper tackles solving polynomial multiparameter eigenvalue problems (PMEPs) by introducing a hidden variable tensor Dixon resultant framework that reduces PMEPs to univariate polynomial eigenvalue problems (PEPs) solvable by standard linearizations. It develops a complete generic-PME solver that leverages hidden-variable resultants and operator-determinant ideas to extract all coordinates of PMEP solutions from PEP eigenpairs, with residual checks to filter spurious results. Theoretical results establish conditions under which the tensor Dixon resultant correctly encodes PMEP solutions for generic systems, while practical strategies address nongeneric cases, including singular pencils and repeated eigenvalues. Numerical experiments in aeroelastic flutter and leaky-wave contexts demonstrate high accuracy and robustness, illustrating the method's broad applicability and potential to replace bespoke linearizations in many PMEP applications.

Abstract

We present a novel, global algorithm for solving polynomial multiparameter eigenvalue problems (PMEPs) by leveraging a hidden variable tensor Dixon resultant framework. Our method transforms a PMEP into one or more univariate polynomial eigenvalue problems, which are solved as generalized eigenvalue problems. Our general approach avoids the need for custom linearizations of PMEPs. We provide rigorous theoretical guarantees for generic PMEPs and give practical strategies for nongeneric systems. Benchmarking on applications from aeroelastic flutter and leaky wave propagation confirms that our algorithm attains high accuracy and robustness while being broadly applicable to many PMEPs.

Paper Structure

This paper contains 25 sections, 4 theorems, 43 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Generic $d$-degree PMEP
  3. Structure of the Paper
  4. Background
  5. The Linear Multiparameter Eigenvalue Problem
  6. Hidden Variable Resultants
  7. The Polynomial Hidden Variable Dixon Resultant
  8. Multivariate Block Vandermonde Vectors
  9. Developing a Resultant for Matrix Polynomials
  10. Linearizations of Polynomial Eigenvalue Problems
  11. A Generic PMEP Solver
  12. The Hidden Variable Tensor Dixon Resultant
  13. Constructing the Dixon function
  14. Extracting Solution Coordinates for Generic PMEPs
  15. Residual Check
  16. ...and 10 more sections

Key Result

Proposition 1

Let $P = \{P_i : 1 \leq i \leq d \}$ be a PMEP as in eq:MPEigForm, expressed in maximal degree $\tau_1,\ldots,\tau_d$ form as in eq:maxdegsystem, where the matrix coefficients of $P_i$ are $n_i \times n_i$ for $n_1,\ldots,n_d \in \mathbb{Z}_{> 0}$. Let $N = \prod_{i=1}^d n_i$, $\alpha_i = i\tau_i with $\mathbf{v} \in \mathbb{C}^{N}$, $x_k^* \in \mathbb{C}, 1 \leq k \leq d-1$. Let $V = \text{vec

Figures (2)

  • Figure 1: Performance of ad hoc linearization and multipareig bor2025rji versus MultiPolyEig graf2025pmep on the system in \ref{['eq:LeakyWaves']}.
  • Figure 2: The connections between methods for four different computational algebra and geometry problems.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Proposition 3
  • proof