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The Schur complements for $SDD_{1}$ matrices and their application to linear complementarity problems

Yang Hu, Jianzhou Liu, Wenlong Zeng

TL;DR

This work addresses the Schur complement behavior of $SDD_{1}$ matrices by introducing a prior construction scaling method tied to the nonnegative inverse-$M$-matrix property. It yields a strictly sharper infinity-norm bound for $A^{-1}$ that depends only on the original entries, and applies this bound to obtain error estimates for linear complementarity problems with $B_{1}$-matrices. Additionally, it provides improved determinant bounds for $SDD_{1}$ matrices and validates the theoretical findings through numerical experiments. The results advance both the theoretical understanding and practical conditioning bounds for problems involving $SDD_{1}$ and related $H$-matrix classes.

Abstract

In this paper we propose a new scaling method to study the Schur complements of $SDD_{1}$ matrices. Its core is related to the non-negative property of the inverse $M$-matrix, while numerically improving the Quotient formula. Based on the Schur complement and a novel norm splitting manner, we establish an upper bound for the infinity norm of the inverse of $SDD_{1}$ matrices, which depends solely on the original matrix entries. We apply the new bound to derive an error bound for linear complementarity problems of $B_{1}$-matrices. Additionally, new lower and upper bounds for the determinant of $SDD_{1}$ matrices are presented. Numerical experiments validate the effectiveness and superiority of our results.

The Schur complements for $SDD_{1}$ matrices and their application to linear complementarity problems

TL;DR

This work addresses the Schur complement behavior of matrices by introducing a prior construction scaling method tied to the nonnegative inverse--matrix property. It yields a strictly sharper infinity-norm bound for that depends only on the original entries, and applies this bound to obtain error estimates for linear complementarity problems with -matrices. Additionally, it provides improved determinant bounds for matrices and validates the theoretical findings through numerical experiments. The results advance both the theoretical understanding and practical conditioning bounds for problems involving and related -matrix classes.

Abstract

In this paper we propose a new scaling method to study the Schur complements of matrices. Its core is related to the non-negative property of the inverse -matrix, while numerically improving the Quotient formula. Based on the Schur complement and a novel norm splitting manner, we establish an upper bound for the infinity norm of the inverse of matrices, which depends solely on the original matrix entries. We apply the new bound to derive an error bound for linear complementarity problems of -matrices. Additionally, new lower and upper bounds for the determinant of matrices are presented. Numerical experiments validate the effectiveness and superiority of our results.

Paper Structure

This paper contains 6 sections, 25 theorems, 167 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Prior construction method for Schur complement
  4. New upper bound of $\left\|A^{-1}\right\|_{\infty}$
  5. The error bound for linear complementarity problems with a $B_{1}$-matrix
  6. The bounds on the determinant for $SDD_{1}$ matrices

Key Result

Lemma 2.1

Varga If $A$ is an $H$-matrix, then

Figures (2)

  • Figure 1: The relation of $N_{1}$, $N_{2}$, $\widetilde{N}_{1}$, $\widetilde{N}_{2}$.
  • Figure 2: $\left\|(I-D+DM)^{-1}\right\|_{\infty}$ for the 5000 matrices $D$ generated by diag(rand(8,1)).

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 49 more