scholarly journals Subdirect Sums of Doubly Strictly Diagonally Dominant Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Yating Li ◽  
Xiaoyong Chen ◽  
Yi Liu ◽  
Lei Gao ◽  
Yaqiang Wang
Keyword(s):  
Sufficient Conditions ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant ◽  
The Given

In this paper, the question of when the subdirect sum of two doubly strictly diagonally dominant (DSDDs) matrices is addressed. Some sufficient conditions are given, and these sufficient conditions only depend on the elements of the given matrices. Moreover, examples are presented to illustrate the corresponding results.

scholarly journals Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices

2015 ◽  
Vol 30 ◽  
pp. 843-870 ◽  
Author(s):  
Cheng-yi Zhang ◽  
Dan Ye ◽  
Cong-Lei Zhong ◽  
SHUANGHUA SHUANGHUA
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Sufficient Conditions ◽  
Numerical Linear Algebra ◽  
Convergence Results ◽  
Diagonally Dominant Matrices ◽  
Sufficient Conditions For Convergence ◽  
Diagonally Dominant ◽  
Necessary And Sufficient ◽  
Theoretical Results

It is well known that as a famous type of iterative methods in numerical linear algebra, Gauss-Seidel iterative methods are convergent for linear systems with strictly or irreducibly diagonally dominant matrices, invertible H−matrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with non-strictly diagonally dominant matrices and general H−matrices. This paper firstly proposes some necessary and sufficient conditions for convergence on Gauss-Seidel iterative methods to establish several new theoretical results on linear systems with nonstrictly diagonally dominant matrices and general H−matrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general H−matrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper.

Simple Criteria for Block H-Matrices

2014 ◽  
Vol 587-589 ◽  
pp. 2307-2311
Author(s):  
Hui Shuang Gao
Keyword(s):  
Necessary Condition ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant ◽  
The Given ◽  
Sufficient And Necessary Condition

In this paper, a sufficient and necessary condition for judging block strictly -diagonally dominant matrices is given firstly. According to the given condition, a new practical criteria for block H-matrices is obtained.

scholarly journals A convergence analysis of SOR iterative methods for linear systems with weak H-matrices

2016 ◽  
Vol 14 (1) ◽  
pp. 747-760
Author(s):  
Cheng-yi Zhang ◽  
Zichen Xue ◽  
Shuanghua Luo
Keyword(s):  
Convergence Analysis ◽  
Linear Systems ◽  
Iterative Methods ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Numerical Examples ◽  
Convergence Results ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant ◽  
Necessary And Sufficient

AbstractIt is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices). However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices). This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.

Practical Criteria for H-Matrices

2014 ◽  
Vol 1049-1050 ◽  
pp. 1316-1319
Author(s):  
Ming Da Zhang
Keyword(s):  
Matrix Theory ◽  
Sufficient Conditions ◽  
Diagonal Dominance ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Strictly Diagonally Dominant Matrix ◽  
Dominant Matrix

Generalized strictly diagonally dominant matrix is very important in computing mathematics and matrix theory, many articles are searching simple and practical identification of generalized strictly diagonally dominant matrix. In this paper, using the concept of diagonal dominance, some sufficient conditions for generalized strictly diagonally dominant matrices are given.

Sufficient Conditions of H-Matrix

2014 ◽  
Vol 651-653 ◽  
pp. 2207-2210
Author(s):  
Ming Da Zhang
Keyword(s):  
Matrix Theory ◽  
Sufficient Conditions ◽  
Diagonal Dominance ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Strictly Diagonally Dominant Matrix ◽  
Dominant Matrix

Generalized strictly diagonally dominant matrix is very important in computing mathematics and matrix theory, many articles are searching simple and practical identification of generalized strictly diagonally dominant matrix. In this paper, using the concept of diagonal dominance, some sufficient conditions for generalized strictly diagonally dominant matrices are given.

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.

Approximating the stationary distribution of an infinite stochastic matrix

1991 ◽  
Vol 28 (1) ◽  
pp. 96-103 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Stationary Distribution ◽  
Numerical Approximation ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Finite System ◽  
Balance Equations ◽  
The Given ◽  
Steady State Balance

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.

scholarly journals Positivity and Monotonicity Preserving Biquartic Rational Interpolation Spline Surface

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Xinru Liu ◽  
Yuanpeng Zhu ◽  
Shengjun Liu
Keyword(s):  
Sufficient Conditions ◽  
Rational Interpolation ◽  
Rectangular Domain ◽  
Interpolation Spline ◽  
Spline Surface ◽  
Surface Data ◽  
Shape Preserving Interpolation ◽  
The Given ◽  
Surface Construction ◽  
Special Case

A biquartic rational interpolation spline surface over rectangular domain is constructed in this paper, which includes the classical bicubic Coons surface as a special case. Sufficient conditions for generating shape preserving interpolation splines for positive or monotonic surface data are deduced. The given numeric experiments show our method can deal with surface construction from positive or monotonic data effectively.

Creative Design Methodology and the SIT Method

1997 ◽  
Author(s):  
Roni Horowitz ◽  
Oded Maimon
Keyword(s):  
Design Methodology ◽  
Search Strategy ◽  
Sufficient Conditions ◽  
Strategy Selection ◽  
Step Procedure ◽  
Creative Solutions ◽  
General Search ◽  
Problem Reformulation ◽  
Creative Problem ◽  
The Given

Abstract The paper presents SIT (Structured Inventive Thinking) — a structured method for enhancing creative problem solving in engineering design. The method is a three step procedure: problem reformulation, general search strategy selection, and an application of idea provoking techniques. The most innovative part of the method is the problem reformulation stage. The given problem is modified through the application of objectively defined and empirically tested set of sufficient conditions for creative solutions. The paper describes the sufficient conditions and the empirical study that demonstrates their appropriateness. Then the whole SIT mechanism is presented with illustrative examples.

scholarly journals Acyclicity Notions for Existential Rules and Their Application to Query Answering in Ontologies

2013 ◽  
Vol 47 ◽  
pp. 741-808 ◽  
Author(s):  
B. Cuenca Grau ◽  
I. Horrocks ◽  
M. Krötzsch ◽  
C. Kupke ◽  
D. Magka ◽  
...  
Keyword(s):  
Knowledge Representation ◽  
Sufficient Conditions ◽  
Query Answering ◽  
Conjunctive Queries ◽  
Practical Solution ◽  
Ontology Language ◽  
Lower Complexity ◽  
The Given ◽  
Theoretical Side

Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a prominent problem in knowledge representation and databases. This problem can be solved using the chase algorithm, which extends the given set of facts with fresh facts in order to satisfy the rules. If the chase terminates, then CQs can be evaluated directly in the resulting set of facts. The chase, however, does not terminate necessarily, and checking whether the chase terminates on a given set of rules and facts is undecidable. Numerous acyclicity notions were proposed as sufficient conditions for chase termination. In this paper, we present two new acyclicity notions called model-faithful acyclicity (MFA) and model-summarising acyclicity (MSA). Furthermore, we investigate the landscape of the known acyclicity notions and establish a complete taxonomy of all notions known to us. Finally, we show that MFA and MSA generalise most of these notions. Existential rules are closely related to the Horn fragments of the OWL 2 ontology language; furthermore, several prominent OWL 2 reasoners implement CQ answering by using the chase to materialise all relevant facts. In order to avoid termination problems, many of these systems handle only the OWL 2 RL profile of OWL 2; furthermore, some systems go beyond OWL 2 RL, but without any termination guarantees. In this paper we also investigate whether various acyclicity notions can provide a principled and practical solution to these problems. On the theoretical side, we show that query answering for acyclic ontologies is of lower complexity than for general ontologies. On the practical side, we show that many of the commonly used OWL 2 ontologies are MSA, and that the number of facts obtained by materialisation is not too large. Our results thus suggest that principled development of materialisation-based OWL 2 reasoners is practically feasible.

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