Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Infinity norm bounds for the inverse of Nekrasov matrices using scaling matrices

Héctor Orera, Juan Manuel Peña

TL;DR

This paper obtains upper bounds for the norm of the inverse of a given matrix when A is a Nekrasov matrix, by means of a scaling matrix transforming A into a strictly diagonally dominant matrix.

Abstract

For many applications, it is convenient to have good upper bounds for the norm of the inverse of a given matrix. In this paper, we obtain such bounds when A is a Nekrasov matrix, by means of a scaling matrix transforming A into a strictly diagonally dominant matrix. Numerical examples and comparisons with other bounds are included. The scaling matrices are also used to derive new error bounds for the linear complementarity problems when the involved matrix is a Nekrasov matrix. These error bounds can improve considerably other previous bounds.

Infinity norm bounds for the inverse of Nekrasov matrices using scaling matrices

TL;DR

This paper obtains upper bounds for the norm of the inverse of a given matrix when A is a Nekrasov matrix, by means of a scaling matrix transforming A into a strictly diagonally dominant matrix.

Abstract

For many applications, it is convenient to have good upper bounds for the norm of the inverse of a given matrix. In this paper, we obtain such bounds when A is a Nekrasov matrix, by means of a scaling matrix transforming A into a strictly diagonally dominant matrix. Numerical examples and comparisons with other bounds are included. The scaling matrices are also used to derive new error bounds for the linear complementarity problems when the involved matrix is a Nekrasov matrix. These error bounds can improve considerably other previous bounds.
Paper Structure (5 sections, 9 theorems, 29 equations, 4 tables, 1 algorithm)

This paper contains 5 sections, 9 theorems, 29 equations, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Scaling matrices
  3. Bounding $\left\Vert{A^{-1}}\right\Vert_\infty$
  4. Improvements, numerical tests and bounds for other norms
  5. Error bounds for LCP of Nekrasov matrices

Key Result

Theorem 2.1

Let $A=(a_{ij})_{1\leq i,j \leq n}$ be a Nekrasov matrix. Then the matrix is a positive diagonal matrix such that $AS$ is SDD.

Theorems & Definitions (17)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Example 4.1
  • Corollary 4.2
  • ...and 7 more