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Condition numbers for the Moore-Penrose inverse and the least squares problem involving rank-structured matrices

Sk. Safique Ahmad, Pinki Khatun

TL;DR

The paper develops a parameter-based perturbation framework to assess structured condition numbers for the Moore–Penrose inverse and the minimum-norm least squares solution when the coefficient matrix is rank-deficient and rank-structured, focusing on Cauchy–Vandermonde and {1,1}-QS matrices. It derives general upper bounds and exact CN expressions under rank-preserving perturbations, and specializes these results to CV and QS matrices, including both QS and GV representations. The authors introduce computationally feasible upper bounds and structured effective CNs that are often much smaller than unstructured CNs, corroborated by extensive numerical experiments. The work enables efficient, structure-aware stability analysis for rank-deficient linear systems and LS problems, with potential extensions to weighted inverses and multi-right-hand-side problems.

Abstract

Perturbation theory plays a crucial role in sensitivity analysis, which is extensively used to assess the robustness of numerical techniques. To quantify the relative sensitivity of any problem, it becomes essential to investigate structured condition numbers (CNs) via componentwise perturbation theory. This paper addresses and analyzes structured mixed condition number (MCN) and componentwise condition number (CCN) for the Moore-Penrose (M-P) inverse and the minimum norm least squares (MNLS) solution involving rank-structured matrices, which include the Cauchy-Vandermonde (CV) matrices and {1, 1}-quasiseparable (QS) matrices. A general framework has been developed to compute the upper bounds for MCN and CCN of rank deficient parameterized matrices. This framework leads to faster computation of upper bounds of structured CNs for CV and {1, 1}-QS matrices. Furthermore, comparisons of obtained upper bounds are investigated theoretically and experimentally. In addition, the structured effective CNs for the M-P inverse and the MNLS solution of {1, 1}-QS matrices are presented. Numerical tests reveal the reliability of the proposed upper bounds as well as demonstrate that the structured effective CNs are computationally less expensive and can be substantially smaller compared to the unstructured CNs.

Condition numbers for the Moore-Penrose inverse and the least squares problem involving rank-structured matrices

TL;DR

The paper develops a parameter-based perturbation framework to assess structured condition numbers for the Moore–Penrose inverse and the minimum-norm least squares solution when the coefficient matrix is rank-deficient and rank-structured, focusing on Cauchy–Vandermonde and {1,1}-QS matrices. It derives general upper bounds and exact CN expressions under rank-preserving perturbations, and specializes these results to CV and QS matrices, including both QS and GV representations. The authors introduce computationally feasible upper bounds and structured effective CNs that are often much smaller than unstructured CNs, corroborated by extensive numerical experiments. The work enables efficient, structure-aware stability analysis for rank-deficient linear systems and LS problems, with potential extensions to weighted inverses and multi-right-hand-side problems.

Abstract

Perturbation theory plays a crucial role in sensitivity analysis, which is extensively used to assess the robustness of numerical techniques. To quantify the relative sensitivity of any problem, it becomes essential to investigate structured condition numbers (CNs) via componentwise perturbation theory. This paper addresses and analyzes structured mixed condition number (MCN) and componentwise condition number (CCN) for the Moore-Penrose (M-P) inverse and the minimum norm least squares (MNLS) solution involving rank-structured matrices, which include the Cauchy-Vandermonde (CV) matrices and {1, 1}-quasiseparable (QS) matrices. A general framework has been developed to compute the upper bounds for MCN and CCN of rank deficient parameterized matrices. This framework leads to faster computation of upper bounds of structured CNs for CV and {1, 1}-QS matrices. Furthermore, comparisons of obtained upper bounds are investigated theoretically and experimentally. In addition, the structured effective CNs for the M-P inverse and the MNLS solution of {1, 1}-QS matrices are presented. Numerical tests reveal the reliability of the proposed upper bounds as well as demonstrate that the structured effective CNs are computationally less expensive and can be substantially smaller compared to the unstructured CNs.
Paper Structure (13 sections, 29 theorems, 68 equations, 10 tables)

This paper contains 13 sections, 29 theorems, 68 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. MCN and CCN for general parameterized matrices
  4. M-P inverse of general parameterized matrices
  5. MNLS solution for general parameterized coefficient matrices
  6. Cauchy-Vandermonde (CV) matrices
  7. Quasiseparable (QS) matrices
  8. CNs corresponding to QS representation
  9. CNs corresponding to GV representation
  10. Comparisons between different CNs for $\{1,1\}$-QS matrices
  11. The structured effective CNs
  12. Numerical experiments
  13. Conclusion

Key Result

Lemma 2.1

wang2018generalized Let $M\in \mathbb{R}^{m\times n}$ with full column rank and $\{E_j\}$ be a collection of real $m \times n$ matrices satisfying ${ \lim_{j\rightarrow 0} }E_j={\bf 0}.$ Then, $(M+E_j)$ has full column rank when $j$ is small enough and ${ \lim _{ j \rightarrow 0}}(M+E_j)^{\dagger}=M

Theorems & Definitions (54)

  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • Remark 1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Definition 3.4
  • Lemma 3.5
  • Theorem 3.6
  • ...and 44 more