Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a perturbation analysis of Higham squared maximum Gaussian elimination growth matrices

Alan Edelman, John Urschel, Bowen Zhu

TL;DR

The paper analyzes the sensitivity of the maximal partial-pivoting growth of Higham$^2$ matrices to entrywise perturbations. It derives explicit perturbation formulas for the last pivot $p^{(i,j)}_\epsilon$ under perturbations $\epsilon \, \mathbf{e}_i \mathbf{e}_j^T$, and provides specialized closed-form expressions for Higham$^2$ matrices, linking perturbation size to conditioning via $\kappa_2(A)$. The results identify highly influential perturbation directions (notably $(1,n-1)$) that can substantially reduce the last pivot and, hence, the growth factor, with numerical experiments corroborating the theory. The findings offer a targeted perturbation framework to improve the stability of Gaussian elimination and shed light on the ridge-like structure of Higham$^2$ matrices in the context of smoothed analysis.

Abstract

Gaussian elimination is the most popular technique for solving a dense linear system. Large errors in this procedure can occur in floating point arithmetic when the matrix's growth factor is large. In the study of numerical linear algebra, it is often valuable to study and characterize the worst case examples. To this end, in their 1989 paper, Higham and Higham characterized the complete set of real n by n matrices that achieves the maximum growth factor under partial pivoting. Left undone is a sensitivity analysis for these matrices under perturbations. The growth factor of these and nearby matrices is the subject of this work. Through theoretical insights and empirical results, we illustrate the high sensitivity of the growth factor of these matrices to perturbations and show how subtle changes can be strategically applied to matrix entries to significantly reduce the growth.

On a perturbation analysis of Higham squared maximum Gaussian elimination growth matrices

TL;DR

The paper analyzes the sensitivity of the maximal partial-pivoting growth of Higham matrices to entrywise perturbations. It derives explicit perturbation formulas for the last pivot under perturbations , and provides specialized closed-form expressions for Higham matrices, linking perturbation size to conditioning via . The results identify highly influential perturbation directions (notably ) that can substantially reduce the last pivot and, hence, the growth factor, with numerical experiments corroborating the theory. The findings offer a targeted perturbation framework to improve the stability of Gaussian elimination and shed light on the ridge-like structure of Higham matrices in the context of smoothed analysis.

Abstract

Gaussian elimination is the most popular technique for solving a dense linear system. Large errors in this procedure can occur in floating point arithmetic when the matrix's growth factor is large. In the study of numerical linear algebra, it is often valuable to study and characterize the worst case examples. To this end, in their 1989 paper, Higham and Higham characterized the complete set of real n by n matrices that achieves the maximum growth factor under partial pivoting. Left undone is a sensitivity analysis for these matrices under perturbations. The growth factor of these and nearby matrices is the subject of this work. Through theoretical insights and empirical results, we illustrate the high sensitivity of the growth factor of these matrices to perturbations and show how subtle changes can be strategically applied to matrix entries to significantly reduce the growth.
Paper Structure (3 sections, 5 theorems, 16 equations, 3 figures)

This paper contains 3 sections, 5 theorems, 16 equations, 3 figures.

Table of Contents

  1. Introduction to Partial Pivoting Growth
  2. Entrywise Perturbations & Higham$^2$ Matrices
  3. Experimental Results

Key Result

Lemma 2.1

Let where $\hat{L} \in \mathrm{SL}_{n-1}(\mathbb{R})$ is lower unitriangular, $\hat{U} \in \mathrm{GL}_{n-1}(\mathbb{R})$ is upper triangular, and $\bm{\ell},\bm{u} \in \mathbb{R}^{n-1}$. Then the LU factorization, if it exists, of $A + \epsilon \, \bm{e}_i \bm{e}_j^T$, where $\bm{e}_i$ is the $i^{th}$ $p^{(i,n)}_\epsilon = p - \epsilon (\bm{\ell}^T \hat{L}^{-1})_i$ for $i<n$, $p^{(n,j)}_\epsil

Figures (3)

  • Figure 1: The effects of an $\epsilon = 10^{-8}$ perturbation at different entries of three $100 \times 100$ Higham$^2$ matrices for different choices of $\hat{U}$ ($\hat{U} = I$ is the Wilkinson matrix). We plot a heatmap for each, where the $(i, j)^{th}$ grid point is the value of $\log_{10}|p^{(i,j)}_\epsilon|$ (in exact arithmetic).
  • Figure 2: Solving $Ax =b$ for Higham$^2$ matrices using Gaussian elimination with no pivoting (GENP) in double precision. For each value of $n$, we generate a Higham$^2$ matrix $A$ (in exact precision), a Gaussian vector $x$, compute $b = Ax$, and solve $A x =b$ for $x$ using GENP applied either to $A$ or $A+ 10^{-8} n B$, where $B$ has independent standard normal entries in either the first row only or the entire matrix. The three scatter plots report the log relative error for each choice of $\hat{U}$ and perturbation method.
  • Figure 3:

Theorems & Definitions (9)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • proof