The stationary iterations revisited

Xuzhou Chen; Xinghua Shi; Yimin Wei

doi:10.3934/naco.2013.3.261

Article Contents

2013, Volume 3, Issue 2: 261-270. Doi: 10.3934/naco.2013.3.261

This issue Previous Article Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming Next Article Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in $\mathbb{R}^n $

The stationary iterations revisited

1.
Department of Computer Science, Fitchburg State University, Fitchburg, MA 01420
2.
Institute of Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433
3.
School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai 200433

Received: January 31, 2012

Revised: December 31, 2012

Published: March 2013

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Abstract

In this paper, we first present a necessary and sufficient conditions for the weakly and strongly convergence of the general stationary iterations $x^{(k+1)} = T x^{(k)} +c$ with initial iteration matrix $T$ and vectors $c$ and $x^{(0)}$. Then we apply these general results and present convergence conditions for the stationary iterations for solving singular linear system $A x = b$. We show that our convergence conditions are weaker and more general than the known results.

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Mathematics Subject Classification: Primary: 65F10, 65F15; Secondary: 15A09.

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