Iterative Method for Linear System with Coefficient Matrix as an $$M_\vee $$ M ∨ -matrix

2017 ◽  
pp. 241-251
Author(s):  
Manideepa Saha
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Coefficient Matrix

Convergence of the Preconditioned AOR Method for an H-Matrix

2013 ◽  
Vol 756-759 ◽  
pp. 2615-2619
Author(s):  
Jie Jing Liu
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Triangular Matrix ◽  
Coefficient Matrix ◽  
Aor Method ◽  
Upper Triangular Matrix ◽  
Theoretical Results

Linear system with H-matrix often appears in a wide variety of areas and is studied by many numerical researchers. In order to improve the convergence rates of iterative method solving the linear system whose coefficient matrix is an H-matrix. In this paper, a preconditioned AOR iterative method with a multi-parameters preconditioner with a general upper triangular matrix is proposed. In addition, the convergence of the coressponding iterative method are established. Lastly, we provide numerical experiments to illustrate the theoretical results.

A Preconditioned AOR Iterative Method for an H-Matrix

2014 ◽  
Vol 644-650 ◽  
pp. 1984-1987
Author(s):  
Shi Guang Zhang
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Triangular Matrix ◽  
Coefficient Matrix ◽  
Numerical Example ◽  
Upper Triangular Matrix

The paper presents a preconditioned AOR iterative method if preconditioner is a general upper triangular matrix for solving a linear system whose coefficient matrix is an H-matrix. In addition, we discuss the convergence of corresponding methods. Finally, a numerical example is also given to illustrate our results.

New Convergence of Relaxed Parallel Multisplitting Method for the H-Matrix

2013 ◽  
Vol 756-759 ◽  
pp. 3162-3166
Author(s):  
You Lin Zhang ◽  
Li Tao Zhang
Keyword(s):  
Linear System ◽  
Theoretical Analysis ◽  
Convergence Rate ◽  
Iterative Method ◽  
Coefficient Matrix ◽  
Convergence Domain ◽  
Multisplitting Method ◽  
Multisplitting Methods ◽  
Relaxed Multisplitting Method

Relaxed technique is one of the main techniques for Improving convergence rate of splitting iterative method. Based on existing parallel multisplitting methods, we have deeply studied the convergence of the relaxed multisplitting method associated with TOR multisplitting for solving the linear system whose coefficient matrix is an H-matrix. Moreover, theoretical analysis have shown that the convergence domain of the relaxed parameters is weaker and wider.

scholarly journals The Preconditioned SOR Iterative Method for Positive Definite Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Yu-Qin Bai ◽  
Yan-Ping Xiao ◽  
Wei-Yuan Ma
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Coefficient Matrix ◽  
Positive Definite ◽  
Positive Definite Matrices ◽  
Numerical Examples

We present several iterations for preconditioners introduced by Tarazaga and Cuellar (2009), and study the convergence of the method for solving a linear system whose coefficient matrix is positive definite matrices, and we also find that they complete very well with the SOR iteration, which is shown through numerical examples.

scholarly journals A Modification of Minimal Residual Iterative Method to Solve Linear Systems

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Xingping Sheng ◽  
Youfeng Su ◽  
Guoliang Chen
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Linear Systems ◽  
Residual Error ◽  
Projection Technique ◽  
Numerical Example ◽  
Minimal Residual

We give a modification of minimal residual iteration (MR), which is 1V-DSMR to solve the linear systemAx=b. By analyzing, we find the modifiable iteration to be a projection technique; moreover, the modification of which gives a better (at least the same) reduction of the residual error than MR. In the end, a numerical example is given to demonstrate the reduction of the residual error between the 1V-DSMR and MR.

scholarly journals Solutions of reaction-diffusion equations using similarity reduction and HSSOR iteration

2019 ◽  
Vol 16 (3) ◽  
pp. 1430
Author(s):  
Nur Afza Mat Ali ◽  
Rostang Rahman ◽  
Jumat Sulaiman ◽  
Khadizah Ghazali
Keyword(s):  
Differential Equation ◽  
Linear System ◽  
Iterative Method ◽  
Iterative Methods ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Sparse Linear System ◽  
Successive Over Relaxation ◽  
Number Of Iterations

<p>Similarity method is used in finding the solutions of partial differential equation (PDE) in reduction to the corresponding ordinary differential equation (ODE) which are not easily integrable in terms of elementary or tabulated functions. Then, the Half-Sweep Successive Over-Relaxation (HSSOR) iterative method is applied in solving the sparse linear system which is generated from the discretization process of the corresponding second order ODEs with Dirichlet boundary conditions. Basically, this ODEs has been constructed from one-dimensional reaction-diffusion equations by using wave variable transformation. Having a large-scale and sparse linear system, we conduct the performances analysis of three iterative methods such as Full-sweep Gauss-Seidel (FSGS), Full-sweep Successive Over-Relaxation (FSSOR) and HSSOR iterative methods to examine the effectiveness of their computational cost. Therefore, four examples of these problems were tested to observe the performance of the proposed iterative methods.  Throughout implementation of numerical experiments, three parameters have been considered which are number of iterations, execution time and maximum absolute error. According to the numerical results, the HSSOR method is the most efficient iterative method in solving the proposed problem with the least number of iterations and execution time followed by FSSOR and FSGS iterative methods.</p>

An optimum iterative method for solving any linear system with a square matrix

1988 ◽  
Vol 28 (1) ◽  
pp. 163-178 ◽  
Author(s):  
Dennis C. Smolarski ◽  
Paul E. Saylor
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Square Matrix

scholarly journals H-Matrices in Fuzzy Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
H. Saberi Najafi ◽  
S. A. Edalatpanah
Keyword(s):  
Linear System ◽  
Theoretical Analysis ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Coefficient Matrix ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Fuzzy Linear System ◽  
Fuzzy Linear Systems

We consider a class of fuzzy linear system of equations and demonstrate some of the existing challenges. Furthermore, we explain the efficiency of this model when the coefficient matrix is an H-matrix. Numerical experiments are illustrated to show the applicability of the theoretical analysis.

scholarly journals An Iterative Method Using Piecewise Valued Basic Vectors for Solving Linear Systems with Symmetric Coefficient Matrix

2004 ◽  
Vol 7 ◽  
pp. 247-254
Author(s):  
Takuro KATAYAMA ◽  
Mitsuhiro KASHIWAGI ◽  
Shin-ichi OHWAKI ◽  
Toshitaka YAMAO
Keyword(s):  
Iterative Method ◽  
Linear Systems ◽  
Coefficient Matrix

scholarly journals Identification of Coupled Map Lattice Based on Compressed Sensing

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Dong Xie ◽  
Lixiang Li ◽  
Xinxin Niu ◽  
Yixian Yang
Keyword(s):  
Linear System ◽  
Compressed Sensing ◽  
Parameter Identification ◽  
Coupling Parameter ◽  
Identification Problem ◽  
Coefficient Matrix ◽  
Coupled Map Lattice ◽  
Theoretical Point ◽  
Mutual Coherence ◽  
Sensing Matrix

A novel approach for the parameter identification of coupled map lattice (CML) based on compressed sensing is presented in this paper. We establish a meaningful connection between these two seemingly unrelated study topics and identify the weighted parameters using the relevant recovery algorithms in compressed sensing. Specifically, we first transform the parameter identification problem of CML into the sparse recovery problem of underdetermined linear system. In fact, compressed sensing provides a feasible method to solve underdetermined linear system if the sensing matrix satisfies some suitable conditions, such as restricted isometry property (RIP) and mutual coherence. Then we give a low bound on the mutual coherence of the coefficient matrix generated by the observed values of CML and also prove that it satisfies the RIP from a theoretical point of view. If the weighted vector of each element is sparse in the CML system, our proposed approach can recover all the weighted parameters using only aboutMsamplings, which is far less than the number of the lattice elementsN. Another important and significant advantage is that if the observed data are contaminated with some types of noises, our approach is still effective. In the simulations, we mainly show the effects of coupling parameter and noise on the recovery rate.

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