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Computing the spectral decomposition of interval matrices and a study on interval matrix power

David Hartman, Milan Hladík, David Říha

TL;DR

This work extends spectral decomposition to interval matrices by enclosing eigenvalues and eigenvectors to obtain an interval spectral decomposition, enabling a robust representation $A=V\Lambda V^{-1}$ (and $A^S=Q\Lambda Q^T$ for symmetric cases) across all realizations. The Bauer--Fike theorem is leveraged to bound eigenvalues within discs around center eigenvalues, while eigenvectors are enclosed via interval linear-system solves, with special handling for symmetric and circulant structures. A numerical study compares binary exponentiation and spectral-decomposition-based power computation across general, symmetric, and special interval matrices, revealing that the spectral approach becomes more efficient for larger powers and benefits greatly from structure (e.g., circulant matrices) or large radii under specific strategies. The results underscore the potential of spectral methods for interval computations, while pointing to improvements from parameterized eigenvector approaches and affine arithmetic to tighten enclosures and broaden applicability. The findings advocate further exploration of structured matrices and advanced interval techniques to enhance performance and reliability in interval linear algebra tasks.

Abstract

We present an algorithm for computing a spectral decomposition of an interval matrix as an enclosure of spectral decompositions of particular realizations of interval matrices. The algorithm relies on tight outer estimations of eigenvalues and eigenvectors of corresponding interval matrices. We present a method for general interval matrices as well as its modification for symmetric interval matrices. As an illustration, we apply the spectral decomposition to computing powers of interval matrices. Numerical results suggest that a simple binary exponentiation is more efficient for smaller exponents, but our approach becomes better when computing higher powers or powers of a special type of matrices. In particular, we consider symmetric interval and circulant interval matrices. In both cases we utilize some properties of the corresponding classes of matrices to make the power computation more efficient.

Computing the spectral decomposition of interval matrices and a study on interval matrix power

TL;DR

This work extends spectral decomposition to interval matrices by enclosing eigenvalues and eigenvectors to obtain an interval spectral decomposition, enabling a robust representation (and for symmetric cases) across all realizations. The Bauer--Fike theorem is leveraged to bound eigenvalues within discs around center eigenvalues, while eigenvectors are enclosed via interval linear-system solves, with special handling for symmetric and circulant structures. A numerical study compares binary exponentiation and spectral-decomposition-based power computation across general, symmetric, and special interval matrices, revealing that the spectral approach becomes more efficient for larger powers and benefits greatly from structure (e.g., circulant matrices) or large radii under specific strategies. The results underscore the potential of spectral methods for interval computations, while pointing to improvements from parameterized eigenvector approaches and affine arithmetic to tighten enclosures and broaden applicability. The findings advocate further exploration of structured matrices and advanced interval techniques to enhance performance and reliability in interval linear algebra tasks.

Abstract

We present an algorithm for computing a spectral decomposition of an interval matrix as an enclosure of spectral decompositions of particular realizations of interval matrices. The algorithm relies on tight outer estimations of eigenvalues and eigenvectors of corresponding interval matrices. We present a method for general interval matrices as well as its modification for symmetric interval matrices. As an illustration, we apply the spectral decomposition to computing powers of interval matrices. Numerical results suggest that a simple binary exponentiation is more efficient for smaller exponents, but our approach becomes better when computing higher powers or powers of a special type of matrices. In particular, we consider symmetric interval and circulant interval matrices. In both cases we utilize some properties of the corresponding classes of matrices to make the power computation more efficient.

Paper Structure

This paper contains 16 sections, 2 theorems, 19 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Interval notation
  3. Spectral decomposition of an interval matrix
  4. Enclosing the eigenvalues
  5. Enclosing the eigenvectors
  6. General case
  7. Symmetric case
  8. Numerical study: Computing powers of interval matrices
  9. General testing pipeline
  10. Powers of general interval matrices
  11. Powers of symmetric interval matrices
  12. Powers of special interval matrices
  13. Circulant matrices
  14. Symmetric interval matrices with large intervals
  15. Discussion of the results
  16. ...and 1 more sections

Key Result

Theorem 1

Let $A,B\in{\mathbb{R}}^{n \times n}$ and suppose that $A$ is diagonalizable, that is, $V^{-1}AV=\mathop{\mathrm{diag}}\nolimits(\lambda_1,\dots,\lambda_n)$ for some $V\in{\mathbb{C}}^{n \times n}$ and $\lambda_1,\dots,\lambda_n\in{\mathbb{C}}$. Then for every (complex) eigenvalue $\lambda(A+B)$ of

Figures (3)

  • Figure 1: (General case) Comparison of the spectral and the binary method for computing interval matrix powers using the ratio \ref{['eq:rate-sum-radii']}. The results were obtained for a testing set of $N_t = 1000$ square matrices of sizes $n \in \{5, 10\}$ with the center $c = 10$ and the radius $r \in \{0.01, 0.001\}$. The light gray line shows the value $1$, representing similar accuracy of the methods.
  • Figure 2: (Symmetric matrices) Comparison of the spectral and the binary method for computing interval matrix powers using the ratio \ref{['eq:rate-sum-radii']}. The results were obtained for a testing set of $N_t = 1000$ square matrices of sizes $n \in \{5, 20\}$ with the center $c = 10$ and the radius $r \in \{0.01, 0.001\}$. The light gray line shows the value $1$, representing similar accuracy of the methods.
  • Figure 3: (Special matrices) Comparison of the spectral and the binary method when computing interval matrix powers using the ratio \ref{['eq:rate-sum-radii']}. (Left) The results shown were obtained for a testing set of $N_t = 1000$ random square circulant matrices of sizes $n = 5$ with the center $c = 10$ and the radius $r = 0.001$. (Right) The results shown were obtained for a testing set of $N_t = 1000$ random square symmetric matrices of sizes $n = 5$ with the center $c = 10$ and the radius $r = 1.0$. The light gray line shows the value $1$, representing similar accuracy of the methods.

Theorems & Definitions (4)

  • Theorem 1: Bauer--Fike, 1960
  • Example 2
  • Proposition 3
  • proof