Modified Successive Overrelaxation Research Articles

Structured Linear Systems and Their Iterative Solutions Through Fuzzy Poisson's Equation

A realistic version of the modified successive overrelaxation (MSOR) with four relaxation parameters is introduced (MMSOR) with application to a representative matrix partition. The one-dimensional Poisson’s equation with fuzzy boundary values is the standard source problem for our treatment (it is sufficient to introduce all the concepts in a simple form). The finite difference method with RedBlack (RB)-Labelling of the grid points is used to introduce a fuzzy algebraic system with characterized fuzzy weak solutions (corresponding to black grid points). We introduce the algorithmic structure and the implementation of MMSOR on the de-fuzzified linear system. The choice of relaxation parameters is based on the minimum Spectral Radius (SR) of the iteration matrices. A comparison with SOR (one relaxation parameter) and MSOR (two relaxation parameters) is considered, and a relation between the three methods is revealed. Assuming the same accuracy, the experimental results showed that the MMSOR runs faster than the SOR and the MSOR methods.

Accelerated Red-Black Strategy for Image Composition Using Laplacian Operator

Digital image composition deals with the problem of embedding portion of source image to the target image to generate a single desirable image seamlessly. The aim is to produce a new image that combine both source and target images so that the seam between the two images is less noticeable. Image composition based on numerical differentiation using Laplacian operator to obtain the solution of Poisson equation is presented. The proposed method employs red-black strategy to speed up the computation by using an accelerated parameter to the existing relaxation method. This modified variant of relaxation method known as Modified Accelerated Over-Relaxation (MAOR) method is derived from the existing Modified Successive Over-Relaxation (MSOR) method. Several examples were tested to examine the effectiveness of the proposed method. The results showed that both modified accelerated variants performed faster than their corresponding standard modified successive variants.

An experimental study of the modified accelerated overrelaxation (MAOR) scheme on stationary helmholtz equation

This research aims to experiment the Modified Accelerated Overrelaxation (MAOR) scheme on second order iterative method for solving two dimensional (2D) Helmholtz Equation. The equation is discretized using the standard second order (Full Sweep) finite difference method. Previous well known relaxation schemes includes the Gauss-Seidel (GS), Successive Overrelaxation (SOR), Modified SOR (MSOR) and Accelerated Overrelaxation (AOR) schemes. These schemes has at most two relaxation parameter while the MAOR scheme have more than two. Several numerical experiments were conducted on two different equations to test the feasibility and the superiority of the MAOR scheme compared to the previous schems on different mesh size.

Numerical assessment for Poisson image blending problem using MSOR iteration via five-point Laplacian operator

The demand for image editing in the field of image processing has been increased throughout the world. One of the most famous equations for solving image editing problem is Poisson equation. Due to the advantages of the Successive Over Relaxation (SOR) iterative method with one weighted parameter, this paper examined the efficiency of the Modified Successive Over Relaxation (MSOR) iterative method for solving Poisson image blending problem. As we know, this iterative method requires two weighted parameters by considering the Red-Black ordering strategy, thus comparison of Jacobi, Gauss-Seidel and MSOR iterative methods in solving Poisson image blending problem is carried out in this study. The performance of these iterative methods to solve the problem is examined through assessing the number of iterations and computational time taken. Based on the numerical assessment over several experiments, the findings had shown that MSOR iterative method is able to solve Poisson image blending problem effectively than the other two methods which it requires fewer number of iterations and lesser computational time.

Boundary Value Problems on Triangular Domains and MKSOR Methods

The performance of six variants of the successive overrelaxation methods (SOR) are considered for an algebraic system arising from a finite difference treatment of an elliptic equation of Partial Differential Equations (PDEs) on a triangular region. The consistency of the finite difference representation of the system is achieved. In the finite difference method one obtains an algebraic system corresponding to the boundary value problem (BVP). The block structure of the algebraic system corresponding to four different labeling (the natural, the red- black and green (RBG), the electronic and the spiral) of the grid points is considered. Also, algebraic systems obtained from BVP with mixed derivatives are well established. Determination of the optimal relaxation parameters on the bases of the graphical representation of the spectral radius of the iteration matrices for the SOR, the Modified Successive over relaxation (MSOR) and their new variants KSOR, MKSOR, MKSOR1 and MKSOR2 are considered. Application of the treatment to two numerical examples is considered.

On the modified successive overrelaxation method

New forms of the modified successive overrelaxation (MSOR); MKSOR, MKSOR1 and MKSOR2 are introduced. These forms depend on the updated successive overrelaxation method (KSOR) and the combination of the SOR and the KSOR methods. We present analysis of the convergence of the MKSOR methods with fixed parameters for solving linear system of equations AX=b, where A belongs to a subclass of consistently ordered matrices. These matrices appear in the numerical treatment of elliptic partial differential equations corresponding to the red black ordering. Also, functional relations for the eigenvalues of the iteration matrices of the Jacobi, the MKSOR methods and the relaxation parameters are established. A general representative numerical example illustrating this treatment is considered. Different values of the relaxation parameters are used on each set of equations. We have implemented our treatment using computer algebra software “Mathematica 8.0”. New results on this subject are presented and compared with those of Young. This approach is promising and will help in the numerical treatment of boundary value problems. Other extensions and applications for further work are mentioned.

The parallel local modified sor for nonsymmetric linear systems*

In this paper we introduce the local Modified Successive Overrelaxation (MSOR) method and apply Fourier analysis to study its convergence. Parallelism is introduced by decoupling the mesh points with the use of red-black ordering for the 5-point stencil. The optimum set of values for the parameters involved, when the Jacobi iteration operator possesses imaginary eigenvalues, is determined. The performance of the local MSOR method is illustrated by its application to the numerical solution of the convection diffusion equation. It is found that the proposed method is significantly more efficient than local SOR when the absolute value of the smallest eigenvalue of the Jacobi operator is larger than unity. Finally, the parallel implementation of the local MSOR method is discussed and results are presented for distributed memory processors with a mesh topology.

Optimal 2-cyclic MSOR for “bowtie” spectra and the “continuous” manteuffel algorithm

For the numerical solution of a linear system whose matrix coefficient is block 2-cyclic consistently ordered, with the eigenvalues of the associated block Jacobi matrix lying in a ‘bowtie” region, several efficient stationary iterative methods have been proposed—among others, by Chin and Manteuffel (1988), Elman and Golub (1990), de Pillis (1991) and Eiermann, Niethammer, and Varga (1992). We propose as an alternative the stationary modified successive overrelaxation (MSOR) method or an “equivalent” two-step method applied to the cyclically reduced linear system. It is shown both theoretically and experimentally that the application of a “continuous” version of Manteuffel's algorithm to derive the optimal parameters of the two-step method produces an iterative method that is asymptotically faster than all the aforementioned methods.

Optimal 2-cyclic MSOR for “bowtie” spectra and the “continuous” Manteuffel algorithm

For the numerical solution of a linear system whose matrix coefficient is block 2-cyclic consistently ordered, with the eigenvalues of the associated block Jacobi matrix lying in a ‘bowtie” region, several efficient stationary iterative methods have been proposed—among others, by Chin and Manteuffel (1988) , Elman and Golub (1990) , de Pillis (1991) and Eiermann, Niethammer, and Varga (1992) . We propose as an alternative the stationary modified successive overrelaxation (MSOR) method or an “equivalent” two-step method applied to the cyclically reduced linear system. It is shown both theoretically and experimentally that the application of a “continuous” version of Manteuffel's algorithm to derive the optimal parameters of the two-step method produces an iterative method that is asymptotically faster than all the aforementioned methods.

On preconditioned msor iterations†

In this paper we prove that, under certain assumptions, the Modified Successive Overrelaxation (MSOR) method applied to a preconditioned linear systems gives improvements in the rate of convergence for some classes of matrices.

An error bound for the modified successive overrelaxation method

In this paper we consider the modified successive overrelaxation (MSOR) method to approximate the solution of the linear system D -1/2 Ax = D -1/2 b, where A is a symmetric, positive definite and consistently ordered matrix and D is a diagonal matrix with the diagonal identical to that of A. The main purpose of this paper is to obtain some theoretical results, namely a bound for the norm of e n = υ - υ n in terms of the norms of δ n = υ n - υ n-1 , δ n+1 = υ n+1 - υ n and their inner product, where υ = D 1/2 x and υ n is the nth iteration vector, obtained using the (MSOR) method.

On a matrix identity connecting iteration operators associated with a p-cyclic matrix

The successive overrelaxation (SOR) and the symmetric SOR (SSOR) iteration matrices are connected with the Jacobi iteration matrix in case these operators are associated with a ( q, p − q)-generalized consistently ordered matrix through certain matrix identities. The validity of these identities has been proved in the last couple of years. Very recently an analogous matrix identity was shown to hold for the modified SOR (MSOR) and the Jacobi iteration matrices in the particular cases ( p,q) = (2,1), (3,1), and 3,2). It is the main objective of this paper to extend the validity of this identity to cover an entire class or pairs ( p, q). The identity in question is not only of theoretical interest but of practical importance too, since it can be used to show the equivalence of the MSOR and a class of p-step iterative methods for the solution of a linear system whose matrix coefficient possesses the ( q,p − q)-generalized consistently ordered property.

On the Convergence of the MSOR Method for Some Classes of Matrices

This paper is concerned with the solution of the linear system $Ax = b$ by the modified successive overrelaxation (MSOR) method, if A belongs to some classes of matrices. It is proven that the classes presented here are subclasses of H-matrices. Since, the general conditions for convergence of MSOR method for the class of H-matrices are not always easy to check in practice, some more practical conditions concerning these subclasses of H-matrices are given in this paper.

Modified successive overrelaxation (MSOR) and equivalent 2-step iterative methods for collocation matrices

We consider a class of consistently ordered matrices which arise from the discretization of Boundary Value Problems (BVPs) when the finite-element collocation method with Hermite elements is used. Through a recently derived equivalence relationship for the asymptotic rates of convergence of the Modified Successive Overrelaxation (MSOR) and a certain 2-step iterative method, we determine the optimum values for the parameters of the MSOR method, as it pertains to collocation matrices. A geometrical algorithm, which utilizes “capturing ellipse” arguments, has been successfully used. The fast convergence properties of the optimum MSOR method are revealed after its comparison to several well-known iterative schemes. Numerical examples, which include the solution of Poisson's equation, are used to verify our results.