scholarly journals Convergence of the GAOR Method for One Subclass ofH-Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Guangbin Wang ◽  
Ting Wang
Keyword(s):  
Linear System ◽  
Convergence Theorem ◽  
Coefficient Matrix ◽  
Linear Least Squares ◽  
Numerical Examples ◽  
Linear Least Squares Problem ◽  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Dominant Matrix ◽  
Better Than

We discuss the convergence of the GAOR method to solve linear system which occurred in solving the weighted linear least squares problem. Moreover, we present one convergence theorem of the GAOR method when the coefficient matrix is a strictly doublyαdiagonally dominant matrix which is a nonsingularH-matrix. Finally, we show that our results are better than previous ones by using four numerical examples.

scholarly journals Improving Results on Convergence of AOR Method

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Guangbin Wang ◽  
Ting Wang ◽  
Yanli Du
Keyword(s):  
Sufficient Conditions ◽  
Numerical Examples ◽  
Diagonally Dominant Matrix ◽  
Aor Method ◽  
Diagonally Dominant ◽  
Dominant Matrix

We present some sufficient conditions on convergence of AOR method for solvingAx=bwithAbeing a strictly doublyαdiagonally dominant matrix. Moreover, we give two numerical examples to show the advantage of the new results.

scholarly journals Criteria for H-Matrices Based on γ−Diagonally Dominant Matrix

2019 ◽  
Vol 11 (6) ◽  
pp. 1
Author(s):  
Xin Li ◽  
Mei Qin
Keyword(s):  
Practical Method ◽  
Diagonal Element ◽  
Principal Diagonal ◽  
Numerical Examples ◽  
Diagonally Dominant Matrix ◽  
The Matrix ◽  
Diagonally Dominant ◽  
Dominant Matrix ◽  
New Criteria

In this paper, we present a new practical criteria for H-matrix based on γ-diagonally dominant matrix. In order to make the judgment conditions convenient and effective, we give two new definitions, one is called strong and weak diagonally dominant degree, the other is called the sum of non-principal diagonal element for the matrix. Further, we obtain a new practical method for the determination of the H-matrix by combining the properties of γ-diagonally dominant matrix, constructing positive diagonal matrix, and adding the appropriate parameters. Finally, we offer numerical examples to verify the validity of the judgment conditions, corresponding numerical examples compared the new criteria and the existing results are presented to verify the advantages of the new determination method.

scholarly journals Second-refinement of Gauss-Seidel iterative method for solving linear system of equations

2020 ◽  
Vol 13 (1) ◽  
pp. 1-15
Author(s):  
Tesfaye Kebede Enyew ◽  
Gurju Awgichew ◽  
Eshetu Haile ◽  
Gashaye Dessalew Abie
Keyword(s):  
Linear System ◽  
Positive Definite ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Symmetric Positive Definite ◽  
Modified Method ◽  
Seidel Method ◽  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Number Of Iterations

Although large and sparse linear systems can be solved using iterative methods, its number of iterations is relatively large. In this case, we need to modify the existing methods in order to get approximate solutions in a small number of iterations. In this paper, the modified method called second-refinement of Gauss-Seidel method for solving linear system of equations is proposed. The main aim of this study was to minimize the number of iterations, spectral radius and to increase rate of convergence. The method can also be used to solve differential equations where the problem is transformed to system of linear equations with coefficient matrices that are strictly diagonally dominant matrices, symmetric positive definite matrices or M-matrices by using finite difference method. As we have seen in theorem 1and we assured that, if A is strictly diagonally dominant matrix, then the modified method converges to the exact solution. Similarly, in theorem 2 and 3 we proved that, if the coefficient matrices are symmetric positive definite or M-matrices, then the modified method converges. And moreover in theorem 4 we observed that, the convergence of second-refinement of Gauss-Seidel method is faster than Gauss-Seidel and refinement of Gauss-Seidel methods. As indicated in the examples, we demonstrated the efficiency of second-refinement of Gauss-Seidel method better than Gauss-Seidel and refinement of Gauss-Seidel methods.

scholarly journals Refinement of Multiparameters Overrelaxation (RMPOR) Method

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Gashaye Dessalew ◽  
Tesfaye Kebede ◽  
Gurju Awgichew ◽  
Assaye Walelign
Keyword(s):  
Positive Definite Matrix ◽  
Positive Definite ◽  
Symmetric Positive Definite Matrix ◽  
Convergence Properties ◽  
Linear System Of Equations ◽  
Symmetric Positive Definite ◽  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Strictly Diagonally Dominant Matrix ◽  
Dominant Matrix

In this paper, we present refinement of multiparameters overrelaxation (RMPOR) method which is used to solve the linear system of equations. We investigate its convergence properties for different matrices such as strictly diagonally dominant matrix, symmetric positive definite matrix, and M-matrix. The proposed method minimizes the number of iterations as compared with the multiparameter overrelaxation method. Its spectral radius is also minimum. To show the efficiency of the proposed method, we prove some theorems and take some numerical examples.

scholarly journals Convergence of GAOR Iterative Method with Strictly Diagonally Dominant Matrices

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Guangbin Wang ◽  
Hao Wen ◽  
Ting Wang
Keyword(s):  
Iterative Method ◽  
Linear Systems ◽  
Numerical Examples ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant ◽  
Better Than

We discuss the convergence of GAOR method for linear systems with strictly diagonally dominant matrices. Moreover, we show that our results are better than ones of Darvishi and Hessari (2006), Tian et al. (2008) by using three numerical examples.

A Method for Judging Generalized Diagonally Dominant Matrix

2002 ◽  
Vol 79 (7) ◽  
pp. 841-848 ◽  
Author(s):  
Xijuan Guo ◽  
Zhihua Liu Chao Jia ◽  
Chao Jia
Keyword(s):  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Dominant Matrix

scholarly journals A Relaxed Kaczmarz Method for Fuzzy Linear Systems

2021 ◽  
Author(s):  
Ke Wang ◽  
Shijun Zhang ◽  
Shiheng Wang
Keyword(s):  
Linear Systems ◽  
Convergence Theorem ◽  
Iterative Scheme ◽  
Coefficient Matrix ◽  
Systems Of Equations ◽  
Numerical Examples ◽  
Right Hand ◽  
Kaczmarz Method ◽  
Fuzzy Linear Systems ◽  
Linear Systems Of Equations

Abstract A relaxed Kaczmarz method is presented for solving a class of fuzzy linear systems of equations with crisp coefficient matrix and fuzzy right-hand side. The iterative scheme is established and the convergence theorem is provided. Numerical examples show that the method is effective.

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