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.

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 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.

Inverse Eigenvalue Problem for a Class of Upper Triangular Matrices with Linear Relation

2014 ◽  
Vol 668-669 ◽  
pp. 1068-1071
Author(s):  
Zhi Bin Li ◽  
Shuai Li
Keyword(s):  
Eigenvalue Problem ◽  
Linear Relation ◽  
Existence And Uniqueness ◽  
Triangular Matrix ◽  
Inverse Eigenvalue Problem ◽  
Triangular Matrices ◽  
Numerical Example ◽  
Upper Triangular Matrices ◽  
Inverse Eigenvalue ◽  
Upper Triangular Matrix

This paper studies on the eigenvalue[1-5] of the a class of upper triangular matrix with linear relation. It discusses the feature of existence and uniqueness of matrix via two given two characteristic pairs(λ,χ),(μ,γ) . Solutions and expressions are provided under satisfied conditions. The possibilities are exanimated by numerical example.

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.

scholarly journals Some Results on Preconditioned Mixed-Type Splitting Iterative Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Guangbin Wang ◽  
Fuping Tan
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Rate Of Convergence ◽  
Mixed Type ◽  
Comparison Theorems ◽  
Numerical Example

We present a preconditioned mixed-type splitting iterative method for solving the linear system Ax=b, where A is a Z-matrix. And we give some comparison theorems to show that the rate of convergence of the preconditioned mixed-type splitting iterative method is faster than that of the mixed-type splitting iterative method. Finally, we give one numerical example 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.

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 Accurate and Efficient Derivative-Free Three-Phase Power Flow Method for Unbalanced Distribution Networks

Computation ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 61
Author(s):  
Oscar Danilo Montoya ◽  
Juan S. Giraldo ◽  
Luis Fernando Grisales-Noreña ◽  
Harold R. Chamorro ◽  
Lazaro Alvarado-Barrios
Keyword(s):  
Power Flow ◽  
Triangular Matrix ◽  
Distribution Networks ◽  
Flow Method ◽  
Processing Times ◽  
Three Phase ◽  
Derivative Free ◽  
Upper Triangular Matrix ◽  
Power Flow Problem ◽  
Power Flow Method

The power flow problem in three-phase unbalanced distribution networks is addressed in this research using a derivative-free numerical method based on the upper-triangular matrix. The upper-triangular matrix is obtained from the topological connection among nodes of the network (i.e., through a graph-based method). The main advantage of the proposed three-phase power flow method is the possibility of working with single-, two-, and three-phase loads, including Δ- and Y-connections. The Banach fixed-point theorem for loads with Y-connection helps ensure the convergence of the upper-triangular power flow method based an impedance-like equivalent matrix. Numerical results in three-phase systems with 8, 25, and 37 nodes demonstrate the effectiveness and computational efficiency of the proposed three-phase power flow formulation compared to the classical three-phase backward/forward method and the implementation of the power flow problem in the DigSILENT software. Comparisons with the backward/forward method demonstrate that the proposed approach is 47.01%, 47.98%, and 36.96% faster in terms of processing times by employing the same number of iterations as when evaluated in the 8-, 25-, and 37-bus systems, respectively. An application of the Chu-Beasley genetic algorithm using a leader–follower optimization approach is applied to the phase-balancing problem utilizing the proposed power flow in the follower stage. Numerical results present optimal solutions with processing times lower than 5 s, which confirms its applicability in large-scale optimization problems employing embedding master–slave optimization structures.

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