Nonnegative iterations with asymptotically constant coefficients

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Let Ak, k = 0, 1, 2, ..., be a sequence of real nonsingular n × n matrices which converge to a nonsingular matrix A. Suppose that A has exactly one positive eigenvalue λ and there exists a unique nonnegative vector u with properties A u = λ u and {norm of matrix} u {norm of matrix} = 1. Under further additional conditions on the spectrum of A, it is shown that if x0 ≠ 0 and the iteratesxk + 1 = Ak xk, k = 0, 1, 2, ...,are nonnegative, then frac(xk, {norm of matrix} xk {norm of matrix}) converges to u and frac({norm of matrix} xk + 1 {norm of matrix}, {norm of matrix} xk {norm of matrix}) converges to λ as k → ∞.

Original languageEnglish
Pages (from-to)1815-1824
Number of pages10
JournalLinear Algebra and Its Applications
Volume431
Issue number10
DOIs
Publication statusPublished - Oct 15 2009

Fingerprint

Non-negative
Iteration
Norm
Coefficient
Matrix Norm
Converge
Nonsingular or invertible matrix
Property A
Eigenvalue

Keywords

  • Convergence
  • Nonnegative iterations
  • z-Transform

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology
  • Numerical Analysis

Cite this

Nonnegative iterations with asymptotically constant coefficients. / Pituk, M.

In: Linear Algebra and Its Applications, Vol. 431, No. 10, 15.10.2009, p. 1815-1824.

Research output: Contribution to journalArticle

@article{01bc073b16f64b63b317b9a3e56531bc,
title = "Nonnegative iterations with asymptotically constant coefficients",
abstract = "Let Ak, k = 0, 1, 2, ..., be a sequence of real nonsingular n × n matrices which converge to a nonsingular matrix A. Suppose that A has exactly one positive eigenvalue λ and there exists a unique nonnegative vector u with properties A u = λ u and {norm of matrix} u {norm of matrix} = 1. Under further additional conditions on the spectrum of A, it is shown that if x0 ≠ 0 and the iteratesxk + 1 = Ak xk, k = 0, 1, 2, ...,are nonnegative, then frac(xk, {norm of matrix} xk {norm of matrix}) converges to u and frac({norm of matrix} xk + 1 {norm of matrix}, {norm of matrix} xk {norm of matrix}) converges to λ as k → ∞.",
keywords = "Convergence, Nonnegative iterations, z-Transform",
author = "M. Pituk",
year = "2009",
month = "10",
day = "15",
doi = "10.1016/j.laa.2009.06.020",
language = "English",
volume = "431",
pages = "1815--1824",
journal = "Linear Algebra and Its Applications",
issn = "0024-3795",
publisher = "Elsevier Inc.",
number = "10",

}

TY - JOUR

T1 - Nonnegative iterations with asymptotically constant coefficients

AU - Pituk, M.

PY - 2009/10/15

Y1 - 2009/10/15

N2 - Let Ak, k = 0, 1, 2, ..., be a sequence of real nonsingular n × n matrices which converge to a nonsingular matrix A. Suppose that A has exactly one positive eigenvalue λ and there exists a unique nonnegative vector u with properties A u = λ u and {norm of matrix} u {norm of matrix} = 1. Under further additional conditions on the spectrum of A, it is shown that if x0 ≠ 0 and the iteratesxk + 1 = Ak xk, k = 0, 1, 2, ...,are nonnegative, then frac(xk, {norm of matrix} xk {norm of matrix}) converges to u and frac({norm of matrix} xk + 1 {norm of matrix}, {norm of matrix} xk {norm of matrix}) converges to λ as k → ∞.

AB - Let Ak, k = 0, 1, 2, ..., be a sequence of real nonsingular n × n matrices which converge to a nonsingular matrix A. Suppose that A has exactly one positive eigenvalue λ and there exists a unique nonnegative vector u with properties A u = λ u and {norm of matrix} u {norm of matrix} = 1. Under further additional conditions on the spectrum of A, it is shown that if x0 ≠ 0 and the iteratesxk + 1 = Ak xk, k = 0, 1, 2, ...,are nonnegative, then frac(xk, {norm of matrix} xk {norm of matrix}) converges to u and frac({norm of matrix} xk + 1 {norm of matrix}, {norm of matrix} xk {norm of matrix}) converges to λ as k → ∞.

KW - Convergence

KW - Nonnegative iterations

KW - z-Transform

UR - http://www.scopus.com/inward/record.url?scp=69349091594&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69349091594&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2009.06.020

DO - 10.1016/j.laa.2009.06.020

M3 - Article

VL - 431

SP - 1815

EP - 1824

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 10

ER -