Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Structured eigenvalue backward errors of Rosenbrock systems and related $μ$-value problems

Anshul Prajapati, Punit Sharma

TL;DR

This work addresses how to quantify the eigenvalue backward error of the Rosenbrock system matrix $S(z)$ under structure-preserving block perturbations. The authors reformulate the backward-error problem as a structured $\mu$-value problem for a related rectangular matrix, deriving explicit constructions and showing exact results in key partial-perturbation cases. They extend classical results to rectangular-block perturbations and provide computable bounds through convexity of joint numerical ranges, aided by partially isometric representations. Numerical experiments on rational eigenvalue problems validate the approach and demonstrate accurate, computable measures of backward error. Overall, the paper offers a practical framework for stability-aware perturbation analysis of rational system representations with concrete computational tools.

Abstract

In this paper, we compute the structured eigenvalue backward error of a Rosenbrock system matrix $S(z)=\left[\begin{array}{cc} A-zI & B \\ C & P(z) \end{array}\right]$ for a given scalar $λ\in \mathbb C$. We have developed simplified formulas for the structured eigenvalue backward error of the Rosenbrock system matrix, considering both full and partial block perturbations. These formulas involve computing structured $μ$-values of a rectangular matrix under rectangular-block-diagonal perturbations. For the reformulated $μ$-value problem, we provide an explicit expression using partial isometric matrices and also obtain a computable upper bound, which is equal to the $μ$-value when the pertrubation matrix has no more than three blocks at the diagonal. The results are illustrated through numerical experiments.

Structured eigenvalue backward errors of Rosenbrock systems and related $μ$-value problems

TL;DR

This work addresses how to quantify the eigenvalue backward error of the Rosenbrock system matrix under structure-preserving block perturbations. The authors reformulate the backward-error problem as a structured -value problem for a related rectangular matrix, deriving explicit constructions and showing exact results in key partial-perturbation cases. They extend classical results to rectangular-block perturbations and provide computable bounds through convexity of joint numerical ranges, aided by partially isometric representations. Numerical experiments on rational eigenvalue problems validate the approach and demonstrate accurate, computable measures of backward error. Overall, the paper offers a practical framework for stability-aware perturbation analysis of rational system representations with concrete computational tools.

Abstract

In this paper, we compute the structured eigenvalue backward error of a Rosenbrock system matrix for a given scalar . We have developed simplified formulas for the structured eigenvalue backward error of the Rosenbrock system matrix, considering both full and partial block perturbations. These formulas involve computing structured -values of a rectangular matrix under rectangular-block-diagonal perturbations. For the reformulated -value problem, we provide an explicit expression using partial isometric matrices and also obtain a computable upper bound, which is equal to the -value when the pertrubation matrix has no more than three blocks at the diagonal. The results are illustrated through numerical experiments.
Paper Structure (12 sections, 94 equations)

This paper contains 12 sections, 94 equations.

Table of Contents

  1. Introduction
  2. Eigenvalue backward error of Rosenbrock system matrix
  3. Backward error with full perturbation
  4. Backward error with partial perturbation
  5. Perturbing only one block of $S(z)$ at a time
  6. Perturbing any two blocks of $S(z)$ at a time
  7. Perturbing any three blocks of $S(z)$ at a time
  8. Structured $\mu$-values of rectangular matrices
  9. Numerical experiments
  10. Conclusion
  11. Proof of Theorem \ref{['thm:formula1']}
  12. Proof of Theorem \ref{['theorem:mu']}

Theorems & Definitions (5)

  • proof
  • proof
  • proof
  • proof
  • proof