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Backward errors for multiple eigenpairs in structured and unstructured nonlinear eigenvalue problems

Miryam Gnazzo, Leonardo Robol

Abstract

Given a nonlinear matrix-valued function $F(λ)$ and approximate eigenpairs $(λ_i, v_i)$, we discuss how to determine the smallest perturbation $δF$ such that $[F + δF](λ_i) v_i = 0$; we call the distance between the $F$ and $F + δF$ the backward error for this set of approximate eigenpairs. We focus on the case where $F(λ)$ is given as a linear combination of scalar functions multiplying matrix coefficients $F_i$, and the perturbation is done on the matrix coefficients. We provide inexpensive upper bounds, and a way to accurately compute the backward error by means of direct computations or through Riemannian optimization. We also discuss how the backward error can be determined when the $F_i$ have particular structures (such as symmetry, sparsity, or low-rank), and the perturbations are required to preserve them. For special cases (such as for symmetric coefficients), explicit and inexpensive formulas to compute the $δF_i$ are also given.

Backward errors for multiple eigenpairs in structured and unstructured nonlinear eigenvalue problems

Abstract

Given a nonlinear matrix-valued function and approximate eigenpairs , we discuss how to determine the smallest perturbation such that ; we call the distance between the and the backward error for this set of approximate eigenpairs. We focus on the case where is given as a linear combination of scalar functions multiplying matrix coefficients , and the perturbation is done on the matrix coefficients. We provide inexpensive upper bounds, and a way to accurately compute the backward error by means of direct computations or through Riemannian optimization. We also discuss how the backward error can be determined when the have particular structures (such as symmetry, sparsity, or low-rank), and the perturbations are required to preserve them. For special cases (such as for symmetric coefficients), explicit and inexpensive formulas to compute the are also given.
Paper Structure (21 sections, 8 theorems, 83 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 8 theorems, 83 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Backward errors for nonlinear eigenvalue problems
  3. Backward errors for given eigenpairs
  4. Backward errors for the eigenvalues
  5. Explicit upper bounds for the backward errors
  6. Structured nonlinear eigenvalue problems
  7. Structured coefficients in linear subspaces
  8. Invariant pairs
  9. Symmetric backward errors
  10. Nonlinear structures
  11. Numerical experiments
  12. Unstructured tests
  13. The Hadeler problem
  14. The beam problem
  15. Test on randomly generated problems
  16. ...and 6 more sections

Key Result

Theorem 2.3

Let $G, V$ be the following matrices: and denote by $G \odot^T V^T$ the Khatri-Rao transpose product between $G$ and $V^T$. Then the backward error $\eta$ is equal to where we define the matrix In particular $\eta \leq \sigma_{\hat{p}}(G \odot^T V^T)^{-1} \|{R}\|_F$, where $\hat{p}$ is the rank of $G \odot^T V^T$.

Figures (6)

  • Figure 1: Comparison among the upper bounds for the unstructured backward error in Theorem \ref{['th:back_err_with_eigenvectors']} and Lemma \ref{['lem:explicit_upper_bounds']}, applied to the Hadeler problem \ref{['eq:hadeler']}.
  • Figure 2: Comparison of the upper bounds for the unstructured backward error for the beam problem in \ref{['eq:beam']}. On the left: we consider $p=3$ approximated eigenpairs. On the right: we consider $p=10$ approximated eigenpairs.
  • Figure 3: Comparison among the upper bounds for the unstructured backward error for the problem \ref{['eq:random-1']}. On the left: we consider a set of $p=3$ approximate eigenpairs. On the right: we consider a set of $p=10$ approximate eigenpairs.
  • Figure 4: Test the upper bound for the structured backward error in Theorem \ref{['th:struct_back_err_with_eigenpairs']}, for the case of randomly generated sparse matrices in Subsection \ref{['subsubsec:randomly_generated_sparse']}. The bound for unstructured case, which does not hold, is reported for completeness.
  • Figure 5: Comparison between the bounds for structured backward error in Theorem \ref{['th:struct_back_err_with_eigenpairs']} and Corollary \ref{['cor:corollary_bound_symm']}, applied to problem \ref{['eq:random-1']} with symmetric coefficients. For completeness, we report the unstructured bound.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • ...and 14 more