Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Structured condition numbers for a linear function of the solution of the generalized saddle point problem

Sk. Safique Ahmad, Pinki Khatun

TL;DR

This work presents a unified framework for structured normwise, mixed, and componentwise condition numbers of a linear function of the generalized saddle point problem (GSPP) solution. It derives closed-form CN expressions under symmetric, Toeplitz, and general linear structures, and extends to structure-preserving perturbations beyond the unstructured setting. The authors demonstrate that structured CNs yield substantially tighter bounds than unstructured CNs and apply the results to weighted Toeplitz regularized least-squares and Tikhonov regularization problems. Numerical experiments confirm the sharpness and practical relevance of the structured CNs, offering a powerful tool for sensitivity analysis in large-scale structured saddle-point systems.

Abstract

This paper addresses structured normwise, mixed, and componentwise condition numbers (CNs) for a linear function of the solution to the generalized saddle point problem (GSPP). We present a general framework that enables us to measure the structured CNs of the individual components of the solution. Then, we derive their explicit formulae when the input matrices have symmetric, Toeplitz, or some general linear structures. In addition, compact formulae for the unstructured CNs are obtained, which recover previous results on CNs for GSPPs for specific choices of the linear function. Furthermore, applications of the derived structured CNs are provided to determine the structured CNs for the weighted Toeplitz regularized least-squares problems and Tikhonov regularization problems, which retrieves some previous studies in the literature.

Structured condition numbers for a linear function of the solution of the generalized saddle point problem

TL;DR

This work presents a unified framework for structured normwise, mixed, and componentwise condition numbers of a linear function of the generalized saddle point problem (GSPP) solution. It derives closed-form CN expressions under symmetric, Toeplitz, and general linear structures, and extends to structure-preserving perturbations beyond the unstructured setting. The authors demonstrate that structured CNs yield substantially tighter bounds than unstructured CNs and apply the results to weighted Toeplitz regularized least-squares and Tikhonov regularization problems. Numerical experiments confirm the sharpness and practical relevance of the structured CNs, offering a powerful tool for sensitivity analysis in large-scale structured saddle-point systems.

Abstract

This paper addresses structured normwise, mixed, and componentwise condition numbers (CNs) for a linear function of the solution to the generalized saddle point problem (GSPP). We present a general framework that enables us to measure the structured CNs of the individual components of the solution. Then, we derive their explicit formulae when the input matrices have symmetric, Toeplitz, or some general linear structures. In addition, compact formulae for the unstructured CNs are obtained, which recover previous results on CNs for GSPPs for specific choices of the linear function. Furthermore, applications of the derived structured CNs are provided to determine the structured CNs for the weighted Toeplitz regularized least-squares problems and Tikhonov regularization problems, which retrieves some previous studies in the literature.
Paper Structure (11 sections, 11 theorems, 103 equations, 8 tables)

This paper contains 11 sections, 11 theorems, 103 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Notation
  4. Preliminaries
  5. CNs for a linear function of the solution to the GSPP when $B=C$
  6. Unstructured CNs forlumae
  7. Structured CNs when $A$ is symmetric and $B=C$ is Toeplitz
  8. Structured CNs when $A$ and $D$ have linear structures
  9. Application to WTRLS problems
  10. Numerical experiments
  11. Conclusions

Key Result

Lemma 2.1

DIAO2007Gohberg1993 Under the same hypothesis as of Definition def21, when $\bm{{\varphi}}$ is $Fr\acute{e}chet$ differentiable at $u$, we have where ${\bf d} \bm{{\varphi}}(u)$ denotes the $Fr\acute{e}chet$ derivative of $\bm{{\varphi}}$ at $u.$

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • ...and 31 more