scholarly journals A preconditioned AOR iterative scheme for systems of linear equations with L-matrics

2019 ◽  
Vol 17 (1) ◽  
pp. 1764-1773
Author(s):  
Hongjuan Wang
Keyword(s):  
Convergence Rate ◽  
Iterative Method ◽  
Iterative Scheme ◽  
Linear Equations ◽  
Comparison Theorems ◽  
Sparse Linear Systems ◽  
Important Method ◽  
Systems Of Linear Equations ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

Abstract In this paper we investigate theoretically and numerically the new preconditioned method to accelerate over-relaxation (AOR) and succesive over-relaxation (SOR) schemes, which are used to the large sparse linear systems. The iterative method that is usually measured by the convergence rate is an important method for solving large linear equations, so we focus on the convergence rate of the different preconditioned iterative methods. Our results indicate that the proposed new method is highly effective to improve the convergence rate and it is the best one in three preconditioned methods that are revealed in the comparison theorems and numerical experiment.

scholarly journals A modified AOR-type iterative method for L-matrix linear systems

2007 ◽  
Vol 49 (2) ◽  
pp. 281-292 ◽  
Author(s):  
Shiliang Wu ◽  
Tingzhu Huang
Keyword(s):  
Iterative Method ◽  
Linear Systems ◽  
Iterative Methods ◽  
Convergence Rates ◽  
Comparison Theorems ◽  
Iterative Schemes ◽  
Seidel Method ◽  
Comparison Results ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

AbstractBoth Evans et al. and Li et al. have presented preconditioned methods for linear systems to improve the convergence rates of AOR-type iterative schemes. In this paper, we present a new preconditioner. Some comparison theorems on preconditioned iterative methods for solving L-matrix linear systems are presented. Comparison results and a numerical example show that convergence of the preconditioned Gauss-Seidel method is faster than that of the preconditioned AOR iterative method.

Comparison Results on Preconditioned AOR-Type Iterative Method for M-Matrices

2013 ◽  
Vol 756-759 ◽  
pp. 2629-2633
Author(s):  
Ting Zhou ◽  
Hong Fang Cui
Keyword(s):  
Linear System ◽  
Convergence Rate ◽  
Iterative Method ◽  
Iterative Methods ◽  
Science And Engineering ◽  
Numerical Example ◽  
Comparison Results ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

For solving the linear system, different preconditioned iterative methods have been proposed by many authors. M-matrices appear in many areas of science and engineering. In this paper, we present preconditioned AOR-type iterative method and the SOR-type iterative method with a preconditioner for solving the M-matrices. In addition, the relation between the convergence rate of preconditioned AOR-type iterative method and the parameters are brought to light. Finally, a numerical example is also given to illustrate the results.

scholarly journals Modified Preconditioned GAOR Methods for Systems of Linear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xue-Feng Zhang ◽  
Qun-Fa Cui ◽  
Shi-Liang Wu
Keyword(s):  
Linear System ◽  
Convergence Rate ◽  
Least Squares ◽  
Linear Equations ◽  
Generalized Least Squares ◽  
Numerical Results ◽  
Least Squares Problem ◽  
Comparison Results ◽  
Systems Of Linear Equations ◽  
Better Than

Three kinds of preconditioners are proposed to accelerate the generalized AOR (GAOR) method for the linear system from the generalized least squares problem. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned generalized AOR (PGAOR) methods is better than that of the original GAOR methods. Finally, some numerical results are reported to confirm the validity of the proposed methods.

scholarly journals Accelerating advanced preconditioning methods on hybrid architectures

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Ernesto Dufrechou
Keyword(s):  
Linear Systems ◽  
Large Scale ◽  
Krylov Subspace ◽  
Linear Equations ◽  
Memory Systems ◽  
Sparse Linear Systems ◽  
Data Parallel ◽  
Systems Of Linear Equations ◽  
Sparse Systems ◽  
Computational Bottleneck

Many problems, in diverse areas of science and engineering, involve the solution of largescale sparse systems of linear equations. In most of these scenarios, they are also a computational bottleneck, and therefore their efficient solution on parallel architectureshas motivated a tremendous volume of research.This dissertation targets the use of GPUs to enhance the performance of the solution of sparse linear systems using iterative methods complemented with state-of-the-art preconditioned techniques. In particular, we study ILUPACK, a package for the solution of sparse linear systems via Krylov subspace methods that relies on a modern inverse-based multilevel ILU (incomplete LU) preconditioning technique.We present new data-parallel versions of the preconditioner and the most important solvers contained in the package that significantly improve its performance without affecting its accuracy. Additionally we enhance existing task-parallel versions of ILUPACK for shared- and distributed-memory systems with the inclusion of GPU acceleration. The results obtained show a sensible reduction in the runtime of the methods, as well as the possibility of addressing large-scale problems efficiently.

SEBSM-Based Iterative Method for Solving Large Systems of Linear Equations and Its Applications in Engineering Computation

2016 ◽  
Vol 13 (05) ◽  
pp. 1650024 ◽  
Author(s):  
Jin-Xiu Hu ◽  
Xiao-Wei Gao ◽  
Zhi-Chao Yuan ◽  
Jian Liu ◽  
Shi-Zhang Huang
Keyword(s):  
Iterative Method ◽  
Direct Method ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Relaxation Factor ◽  
Banded Matrix ◽  
Speed Up ◽  
Systems Of Linear Equations ◽  
Matrix Formula ◽  
Large Systems

In this paper, a new iterative method, for solving sparse nonsymmetrical systems of linear equations is proposed based on the Simultaneous Elimination and Back-Substitution Method (SEBSM), and the method is applied to solve systems resulted in engineering problems solved using Finite Element Method (FEM). First, SEBSM is introduced for solving general linear systems using the direct method. And, then an iterative method based on SEBSM is presented. In the method, the coefficient matrix [Formula: see text] is split into lower, diagonally banded and upper matrices. The iterative convergence can be controlled by selecting a suitable bandwidth of the diagonally banded matrix. And the size of the working array needing to be stored in iteration is as small as the bandwidth of the diagonally banded matrix. Finally, an accelerating strategy for this iterative method is proposed by introducing a relaxation factor, which can speed up the convergence effectively if an optimal relaxation factor is chosen. Two numerical examples are given to demonstrate the behavior of the proposed method.

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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