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Rigorous Perturbation Bounds for the QX Decomposition for Centrosymmetric Matrices

Aamir Farooq, Rewayat Khan, Uzma Rani, M. Tariq Rahim

TL;DR

The paper develops a rigorous perturbation analysis for the structure-preserving $QX$ decomposition of centrosymmetric matrices, deriving both norm-wise and component-wise bounds. It employs two analytic frameworks: a refined matrix equation method for weak bounds and a modified matrix-vector approach with Lyapunov majorants and Banach fixed-point theory for strong bounds, complemented by explicit mixed and component-wise condition numbers. Theoretical results are supported by numerical experiments that validate sharpness and provide guidance on when each bound is preferable. This work enhances forward-error estimation and sensitivity analysis for structure-preserving QR-like factorizations in centrosymmetric settings, with potential extensions to multiplicative perturbations.

Abstract

Konrad Burnik suggests a structure-preserving QR factorization for centrosymmetric matrices, known as QX factorization. In this article, we obtain the explicit expressions for rigorous perturbation bounds of the QX factorization when the original matrix is perturbed, either norm-wise or component-wise. First, using the matrix-equation approach, weak rigorous perturbation bounds are derived. Then, strong rigorous perturbation bounds are obtained by combining the modified matrix-vector equation approach with the strategy for the Lyapunov majorant function and the Banach fixed-point theorem. The mixed and component-wise condition numbers and their upper bounds are also explicitly expressed. Numerical tests illustrate the validity of the obtained results.

Rigorous Perturbation Bounds for the QX Decomposition for Centrosymmetric Matrices

TL;DR

The paper develops a rigorous perturbation analysis for the structure-preserving decomposition of centrosymmetric matrices, deriving both norm-wise and component-wise bounds. It employs two analytic frameworks: a refined matrix equation method for weak bounds and a modified matrix-vector approach with Lyapunov majorants and Banach fixed-point theory for strong bounds, complemented by explicit mixed and component-wise condition numbers. Theoretical results are supported by numerical experiments that validate sharpness and provide guidance on when each bound is preferable. This work enhances forward-error estimation and sensitivity analysis for structure-preserving QR-like factorizations in centrosymmetric settings, with potential extensions to multiplicative perturbations.

Abstract

Konrad Burnik suggests a structure-preserving QR factorization for centrosymmetric matrices, known as QX factorization. In this article, we obtain the explicit expressions for rigorous perturbation bounds of the QX factorization when the original matrix is perturbed, either norm-wise or component-wise. First, using the matrix-equation approach, weak rigorous perturbation bounds are derived. Then, strong rigorous perturbation bounds are obtained by combining the modified matrix-vector equation approach with the strategy for the Lyapunov majorant function and the Banach fixed-point theorem. The mixed and component-wise condition numbers and their upper bounds are also explicitly expressed. Numerical tests illustrate the validity of the obtained results.

Paper Structure

This paper contains 9 sections, 3 theorems, 122 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Rigorous Perturbation Bounds
  3. Rigorous Perturbation Bounds By Using Refined Matrix Equation Method
  4. Rigorous Perturbation Bounds by Using Modified Matrix-Vector Equation Approach
  5. Component-wise Perturbation
  6. Condition numbers
  7. Mixed and component-wise condition numbers
  8. Numerical Experiments
  9. Conclusion

Key Result

Lemma 1.1

lv2022perturbation: Let $\mathbb{D}_n$$\subseteq \mathbb{R}^{n\times n}$ be the set of diagonal centrosymmetric matrices with positive diagonal elements and $n=2l$. Then, for any $D_n=\textmd{diag}(\delta_{1},\delta_{2},...,\delta_{n},\delta_{n},...,\delta_{2},\delta_{1})\in$$\mathbb{D}_n$ and centr where $\varsigma_{D_{n}} = \mathop {\max }\limits_{1 \le \alpha < \beta \le n} \{\delta_{\beta}/\de

Theorems & Definitions (20)

  • Lemma 1.1
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Remark 2.2
  • Remark 2.3
  • proof
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 10 more