scholarly journals Refinement of Extended Accelerated Over-Relaxation Method for Solution of Linear Systems

2021 ◽  
Vol 4 (1) ◽  
pp. 53-61
Author(s):  
KJ Audu ◽  
YA Yahaya ◽  
KR Adeboye ◽  
UY Abubakar
Keyword(s):  
Spectral Radius ◽  
Linear Equations ◽  
Relaxation Method ◽  
System Of Linear Equations ◽  
Iteration Matrix ◽  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Dominant Matrix ◽  
Successive Over Relaxation ◽  
Stationary Iterative Methods

Given any linear stationary iterative methods in the form z^(i+1)=Jz^(i)+f, where J is the iteration matrix, a significant improvements of the iteration matrix will decrease the spectral radius and enhances the rate of convergence of the particular method while solving system of linear equations in the form Az=b. This motivates us to refine the Extended Accelerated Over-Relaxation (EAOR) method called Refinement of Extended Accelerated Over-Relaxation (REAOR) so as to accelerate the convergence rate of the method. In this paper, a refinement of Extended Accelerated Over-Relaxation method that would minimize the spectral radius, when compared to EAOR method, is proposed. The method is a 3-parameter generalization of the refinement of Accelerated Over-Relaxation (RAOR) method, refinement of Successive Over-Relaxation (RSOR) method, refinement of Gauss-Seidel (RGS) method and refinement of Jacobi (RJ) method. We investigated the convergence of the method for weak irreducible diagonally dominant matrix, matrix or matrix and presented some numerical examples to check the performance of the method. The results indicate the superiority of the method over some existing methods.

scholarly journals Genetic programming for iterative numerical methods

2021 ◽  
Author(s):  
Dominik Sobania ◽  
Jonas Schmitt ◽  
Harald Köstler ◽  
Franz Rothlauf
Keyword(s):  
Genetic Programming ◽  
Linear Systems ◽  
Spectral Radius ◽  
Linear Equations ◽  
Parallel Computer ◽  
Iteration Matrix ◽  
Mathematical Expressions ◽  
Lower Spectral Radius ◽  
Problem Instances ◽  
Stationary Iterative Methods

AbstractWe introduce GPLS (Genetic Programming for Linear Systems) as a GP system that finds mathematical expressions defining an iteration matrix. Stationary iterative methods use this iteration matrix to solve a system of linear equations numerically. GPLS aims at finding iteration matrices with a low spectral radius and a high sparsity, since these properties ensure a fast error reduction of the numerical solution method and enable the efficient implementation of the methods on parallel computer architectures. We study GPLS for various types of system matrices and find that it easily outperforms classical approaches like the Gauss–Seidel and Jacobi methods. GPLS not only finds iteration matrices for linear systems with a much lower spectral radius, but also iteration matrices for problems where classical approaches fail. Additionally, solutions found by GPLS for small problem instances show also good performance for larger instances of the same problem.

scholarly journals Second Degree Generalized Successive Over Relaxation Method for Solving System of Linear Equations

2020 ◽  
Vol 12 (1) ◽  
pp. 60-71
Author(s):  
Firew Hailu ◽  
Genanew Gofe Gonfa ◽  
Hailu Muleta Chemeda
Keyword(s):  
Iterative Method ◽  
Linear Equations ◽  
Relaxation Method ◽  
The Other ◽  
System Of Linear Equations ◽  
Convergence Properties ◽  
Numerical Examples ◽  
Successive Over Relaxation ◽  
Number Of Iterations ◽  
Second Degree

In this paper, a second degree generalized successive over relaxation iterative method for solving system of linear equations based on the decomposition  A= Dm+Lm+Um  is presented and the convergence properties of the proposed method are discussed. Two numerical examples are considered to show the efficiency of the proposed method. The results presented in tables show that the Second Degree Generalized Successive Over Relaxation Iterative method is more efficient than the other methods considered based on number of iterations, computational running time and accuracy. Keywords: Second Degree, Generalized Gauss Seidel, Successive over relaxation, Convergence.

scholarly journals Second degree generalized Jacobi iteration method for solving system of linear equations

2016 ◽  
Vol 47 (2) ◽  
pp. 179-192
Author(s):  
Tesfaye Kebede Enyew
Keyword(s):  
Spectral Radius ◽  
Iteration Method ◽  
Linear Equations ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Jacobi Iteration ◽  
Optimal Values ◽  
Second Degree

In this paper, a Second degree generalized Jacobi Iteration method for solving system of linear equations, $Ax=b$ and discuss about the optimal values $a_{1}$ and $b_{1}$ in terms of spectral radius about for the convergence of SDGJ method of $x^{(n+1)}=b_{1}[D_{m}^{-1}(L_{m}+U_{m})x^{(n)}+k_{1m}]-a_{1}x^{(n-1)}.$ Few numerical examples are considered to show that the effective of the Second degree Generalized Jacobi Iteration method (SDGJ) in comparison with FDJ, FDGJ, SDJ.

A property for the \alpha-diagonally dominant matrix with applications

2013 ◽  
Vol 7 ◽  
pp. 2111-2116
Author(s):  
Zhi-Jun Guo ◽  
Xu-Hui Shen ◽  
Yu-Ming Chu
Keyword(s):  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Dominant Matrix

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.

Computing the nearest diagonally dominant matrix

1998 ◽  
Vol 5 (6) ◽  
pp. 461-474 ◽  
Author(s):  
María Mendoza ◽  
Marcos Raydan ◽  
Pablo Tarazaga
Keyword(s):  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Dominant Matrix

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 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 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.

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
Sign in / Sign up

Export Citation Format

Share Document