Accelerated Overrelaxation Iterative Method Research Articles

New preconditioning and half-sweep accelerated overrelaxation solution for fractional differential equation

The present paper investigates the approximate solution of a one-dimensional linear space-fractional diffusion equation using a new preconditioning matrix to develop an efficient half-sweep accelerated overrelaxation iterative method. The proposed method utilizes unconditionally stable implicit finite difference schemes to formulate the discrete approximation equation to the problem. The formulation employs the Caputo fractional derivative to treat the space-fractional derivative in the problem. The paper's focus is to assess the improvement in terms of the convergence rate of the solution obtained by the proposed iterative method. The numerical experiment illustrates the superiority of the proposed method in terms of solution efficiency against one of the existing preconditioned methods, preconditioned accelerated overrelaxation and implicit Euler method. The proposed method reveals the ability to compute the solution with lesser iterations and faster computation time than the preconditioned accelerated overrelaxation and implicit Euler method. The method introduced in the paper, half-sweep preconditioned accelerated overrelaxation, has the potential to solve a variety of space-fractional diffusion models efficiently. Future investigation will improve the absolute errors of the solutions.

A low-complexity signal detection utilizing AOR iterative method for massive MIMO systems

Massive multiple-input multiple-output (MIMO) system is capable of substantially improving the spectral efficiency as well as the capacity of wireless networks relying on equipping a large number of antenna elements at the base stations. However, the excessively high computational complexity of the signal detection in massive MIMO systems imposes a significant challenge for practical hardware implementations. In this paper, we propose a novel minimum mean square error (MMSE) signal detection using the accelerated overrelaxation (AOR) iterative method without complicated matrix inversion, which is capable of reducing the overall complexity of the classical MMSE algorithm by an order of magnitude. Simulation results show that the proposed AOR-based method can approach the conventional MMSE signal detection with significant complexity reduction.

Cubic B-spline solution for two-point boundary value problem with AOR iterative method

In this study, the cubic B-spline approximation equation has been derived by using the cubic B-spline discretization scheme to solve two-point boundary value problems. In addition to that, system of cubic B-spline approximation equations is generated from this spline approximation equation in order to get the numerical solutions. To do this, the Accelerated Over Relaxation (AOR) iterative method has been used to solve the generated linear system. For the purpose of comparison, the GS iterative method is designated as a control method to compare between SOR and AOR iterative methods. There are two examples of proposed problems that have been considered to examine the efficiency of these proposed iterative methods via three parameters such as their number of iterations, computational time and maximum absolute error. The numerical results are obtained from these iterative methods, it can be concluded that the AOR iterative method is slightly efficient as compared with SOR iterative method.

Accelerated over relaxation iterative method using triangle element approximation for solving 2D Helmholtz Equations

Weighted iterative methods particularly Accelerated Over Relaxation (AOR) method are used to solve linear system generated from triangle finite element approximation equation in solving 2D Helmholtz equation. The development of the AOR iterative method were also presented. Numerical experiments have been carried out and the results obtained confirm the superiority of the proposed iterative method

A Preconditioned AOR Iterative Method for the Absolute Value Equations

In this paper, coupled with preconditioning technique, a preconditioned accelerated over relaxation (PAOR) iterative method for solving the absolute value equations (AVEs) is presented. Some comparison theorems are given when the matrix of the linear term is an irreducible [Formula: see text]-matrix. Comparison results show that the convergence rate of the PAOR iterative method is better than that of the accelerated over relaxation (AOR) iterative method whenever both are convergent. Numerical experiments are provided in order to confirm the theoretical results studied in this paper.

The solution of the linear complementarity problem by the matrix analogue of the accelerated overrelaxation iterative method

The Linear Complementarity Problem (LCP), with an H+źmatrix coefficient, is solved by using the new "(Projected) Matrix Analogue of the AOR (MAAOR)" iterative method; this new method constitutes an extension of the "Generalized AOR (GAOR)" iterative method. In this work two sets of convergence intervals of the parameters involved are determined by the theories of "Perron-Frobenius" and of "Regular Splittings". It is shown that the intervals in question are better than any similar convergence intervals found so far by similar iterative methods. A deeper analysis reveals that the "best" values of the parameters involved are those of the (projected) scalar Gauss-Seidel iterative method. A theoretical comparison of the "best" (projected) Gauss-Seidel and the "best" modulus-based splitting Gauss-Seidel method is in favor of the former method. A number of numerical examples support most of our theoretical findings.

The matrix analogue of the scalar AOR iterative method

The Accelerated Overrelaxation (AOR) and the Generalized AOR (GAOR) iterative methods for the solution of linear systems of algebraic equations (Ax=b,A∈Cn×n,det(A)≠0,b∈Cn) have been around for about four decades and a plethora of variations of them have been proposed. In this work a novel algorithm is introduced, the Matrix Analogue of the AOR (MAAOR) iterative method, which is analysed and studied. The MAAOR method generalizes both the AOR and the GAOR. Sufficient convergence conditions for the GAOR method are determined when the coefficient matrix A of the linear system to be solved is a Hermitian matrix with positive diagonal elements. Similarly, sufficient convergence conditions for the MAAOR method are determined when A is a nonsingular H-matrix. The new convergence conditions are the most general ones so far. Numerical examples are presented in support of the theory developed.

On the Solution of the Linear Complementarity Problem by the Generalized Accelerated Overrelaxation Iterative Method

In the present work, we determine intervals of convergence for the various parameters involved for what is known as the generalized accelerated overrelaxation iterative method for the solution of the linear complementarity problem. The convergence intervals found constitute sufficient conditions for the generalized accelerated overrelaxation method to converge and are better than what have been known so far.

A New Version of the Accelerated Overrelaxation Iterative Method

Hadjidimos (1978) proposed a classical accelerated overrelaxation (AOR) iterative method to solve the system of linear equations, and discussed its convergence under the conditions that the coefficient matrices are irreducible diagonal dominant,L-matrices, and consistently orders matrices. In this paper, a new version of the AOR method is presented. Some convergence results are derived when the coefficient matrices are irreducible diagonal dominant,H-matrices, symmetric positive definite matrices, andL-matrices. A relational graph for the new AOR method and the original AOR method is presented. Finally, a numerical example is presented to illustrate the efficiency of the proposed method.

A variant of the AOR method for augmented systems

In this paper we present a variant of the Accelerated Overrelaxation iterative method (AOR), denoted by modified AOR-like method (MAOR-like method) for solving the augmented systems, i.e. the AOR-like method with three real parameters , r and . For special values of r ,  and  we get the MSOR-like method, the AOR-like method and the SOR-like method. An equation relating the involved parameters and the eigenvalues of the iteration matrix of the MAOR-like method is obtained. Furthermore, some convergence conditions for the MAOR-like method are derived. This paper generalizes the main results of Li, Li, Nie, and Evans 2004 and Shao, Li, and Li (2007). Numerical examples are presented to show that, for a suitable choice of the involved parameters, the MAOR-like method is superior when compared to the above iterative methods and to the SSOR-like method presented by Zheng, Wang, and Wu (2009).

AOR iterative methods for rank deficient least squares problems

In this paper we study the semiconvergence of accelerated overrelaxation (AOR) iterative methods for the least squares solution of minimal norm of rank deficient linear systems. Necessary and sufficient conditions for the semiconvergence of the AOR and JOR iterative methods are given. The optimum parameters and the associated convergence factor are derived.

A convergence analysis of block accelerated over-relaxation iterative methods for weak block H-matrices to partition π

The aim of this paper is to establish the convergence of the block iteration methods such as the block successively accelerated over-relaxation method (BAOR) and the symmetric block successively accelerated over-relaxation method (BSAOR): Let A ∈ C π , n m , m be a weak block H-matrix to partition π, then for 0 ⩽ r ⩽ ω ⩽ 2 1 + ρ ( | B J ( A ) | ) , ρ ( B L r , ω ) ⩽ | 1 - ω | + ω ρ ( | B J ( A ) | ) , ρ ( B S r , ω ) ⩽ [ | 1 - ω | + ω ρ ( | B J ( A ) | ) ] 2 , and exact convergence and divergence domains for the block SOR and block SSOR iterative methods are obtained as it has been obtained to H-matrices. Based on these results, the main results in Bai [Parallel Computing 25 (1999)] and Cvetković [Appl. Numer. Math. 41 (2002)] can be improved.

Models of asynchronous parallel matrix multisplitting relaxed iterations

By making use of the principle of sufficiently using the delayed information, we propose two models of asynchronous parallel matrix multisplitting accelerated overrelaxation iterative methods for solving systems of linear equations, which have the merits of convenient computations, flexible and free communications, and can cover all the known synchronous as well as asynchronous parallel matrix multisplitting relaxation methods and their special cases. When the coefficient matrix is an H-matrix, we prove the convergence and estimate the convergence rates of these models in a detailed manner.

Convergence conditions for some generalized iterative methods

In this paper we establish some sufficient conditions of convergence for the Generalized Accelerated Overrelaxation iterative method (GAOR), when the matrix A of Ax = b belongs to the class Cl and is an H-matrix

The aor method for AX - XB = C

In this paper we present the block Accelerated Overrelaxation iterative method (AOR) to approximate the solution of the matrix equation AX - XB - C. An analytic determination of good values of the involved real parameters of this iterative method is presented in terms of certain bounds on the eigenvalues of the iteration matrix.

A systolic design for the A.O.R. and S.A.O.R. iterative methods for solving linear systems

In this paper the systolic design for the Accelerated Overrelaxation iterative method [1] for solving large linear systems of linear equations is presented based on the VLSI techniques discussed earlier in Evans and Haider [2] for the Jacobi, Gauss-Seidel, S.O.R. and S.S.O.R. methods. Finally the Conrad-Wallach strategy [3] is used to accomplish the S.A.O.R. method involving no extra computational work.

On an Accelerated Overrelaxation Iterative Method for Linear Systems With Strictly Diagonally Dominant Matrix

We consider a linear system $Ax = b$ of n simultaneous equations, where A is a strictly diagonally dominant matrix. We get bounds for the spectral radius of the matrix ${L_{r,\omega }}$, which is accociated with the Accelerated Overrelaxation iterative method (AOR). Sufficient conditions for the convergence of that method will be given, which improve the results of Theorem 3, Section 4 of [2], applied to this type of matrices.

On an accelerated overrelaxation iterative method for linear systems with strictly diagonally dominant matrix

We consider a linear system A x = b Ax = b of n simultaneous equations, where A is a strictly diagonally dominant matrix. We get bounds for the spectral radius of the matrix L r , ω {L_{r,\omega }} , which is accociated with the Accelerated Overrelaxation iterative method (AOR). Sufficient conditions for the convergence of that method will be given, which improve the results of Theorem 3, Section 4 of [2], applied to this type of matrices.