scholarly journals Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem

1962 ◽  
Vol 12 (4) ◽  
pp. 1241-1250 ◽  
Author(s):  
David G. Feingold ◽  
Richard Varga
Keyword(s):  
Circle Theorem ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant ◽  
Gerschgorin Circle

scholarly journals Polyhedral approximations of the semidefinite cone and their application

2021 ◽  
Author(s):  
Yuzhu Wang ◽  
Akihiro Tanaka ◽  
Akiko Yoshise
Keyword(s):  
Initial Approximation ◽  
Semidefinite Relaxation ◽  
Stable Set ◽  
Cutting Plane Methods ◽  
Diagonally Dominant Matrices ◽  
Maximum Stable Set ◽  
Simple Expansion ◽  
Diagonally Dominant ◽  
Polyhedral Approximations ◽  
Semidefinite Cone

AbstractWe develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

The Schur Complement on Weak Block Diagonally Dominant Matrices and Weak Block H-Matrices

2011 ◽  
Vol 148-149 ◽  
pp. 1523-1526
Author(s):  
Shi Hong Liu ◽  
Hong Su ◽  
Zhuo Hong Huang
Keyword(s):  
Schur Complement ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant

In this paper, we prove that the Schur complement of Weak block diagonally dominant matrices and weak block H-matrices are Weak block diagonally dominant matrices and weak block H-matrices, respectively.

scholarly journals Convergence of Relaxed Matrix Parallel Multisplitting Chaotic Methods forH-Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Li-Tao Zhang ◽  
Jian-Lei Li ◽  
Tong-Xiang Gu ◽  
Xing-Ping Liu
Keyword(s):  
Numerical Experiments ◽  
Convergence Property ◽  
Numerical Examples ◽  
Convergence Results ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant

Based on the methods presented by Song and Yuan (1994), we construct relaxed matrix parallel multisplitting chaotic generalized USAOR-style methods by introducing more relaxed parameters and analyze the convergence of our methods when coefficient matrices areH-matrices or irreducible diagonally dominant matrices. The parameters can be adjusted suitably so that the convergence property of methods can be substantially improved. Furthermore, we further study some applied convergence results of methods to be convenient for carrying out numerical experiments. Finally, we give some numerical examples, which show that our convergence results are applied and easily carried out.

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