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 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>

scholarly journals Approximating the Inverse of a Square Matrix with Application in Computation of the Moore-Penrose Inverse

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
F. Soleymani ◽  
M. Sharifi ◽  
S. Shateyi
Keyword(s):  
Iterative Method ◽  
Numerical Results ◽  
Order Convergence ◽  
Approximate Inverses ◽  
Square Matrix

This paper presents a computational iterative method to find approximate inverses for the inverse of matrices. Analysis of convergence reveals that the method reaches ninth-order convergence. The extension of the proposed iterative method for computing Moore-Penrose inverse is furnished. Numerical results including the comparisons with different existing methods of the same type in the literature will also be presented to manifest the superiority of the new algorithm in finding approximate inverses.

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.

scholarly journals An Iterative Method for Symmetric Positive Semidefinite Linear System of Equations

2014 ◽  
Vol 47 (2) ◽  
Author(s):  
Davod Khojasteh Salkuyeh
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Numerical Experiments ◽  
Positive Semidefinite ◽  
Sufficient Condition ◽  
System Of Equations ◽  
Linear System Of Equations

AbstractIn this paper, a new two-step iterative method for solving symmetric positive semidefinite linear system of equations is presented. A sufficient condition for the semiconvergence of the method is also given. Some numerical experiments are presented to show the efficiency of the proposed method.

scholarly journals Remarks on the convergence of an iterative method of solution of generalized least squares problem

2010 ◽  
Vol 43 (4) ◽  
Author(s):  
Zbigniew Bartoszewski
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Least Squares ◽  
Generalized Least Squares ◽  
General Linear ◽  
Least Squares Problem ◽  
Method Of Solution ◽  
Regularized Solution

AbstractIn this paper we consider an iterative method of finding a regularized solution of a general linear system

Convergence Results of the Improving Modified Gauss-Seidel (IMGS) Iterative Method for a Symmetric Positive Definite Matrix

2014 ◽  
Vol 989-994 ◽  
pp. 1794-1797
Author(s):  
Shi Guang Zhang ◽  
Ting Zhou
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Convergence Rates ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Symmetric Positive Definite Matrix ◽  
Symmetric Positive Definite ◽  
Convergence Results ◽  
Comparison Results

In this paper, in order to improve the convergence rates of iterative method solving the linear system, the improving modified Gauss-Seidel (IMGS) iterative method with a preconditioner is proposed. Some convergence and comparison results are given when is a symmetric definite matrix are provided.

Convergence Analysis of Preconditioned SSOR Iterative Method for Linear Systems

2014 ◽  
Vol 644-650 ◽  
pp. 1988-1991
Author(s):  
Ting Zhou
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Convergence Analysis ◽  
Linear Systems ◽  
Iterative Methods ◽  
Convergence Result ◽  
Numerical Example ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

For solving the linear system, different preconditioned iterative methods have been proposed by many authors. In this paper, we present preconditioned SSOR iterative method for solving the linear systems. Meanwhile, we apply the preconditioner to H-matrix and obtain the convergence result. Finally, a numerical example is also given to illustrate our results.

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