Effectiveness of Preconditioned m-Order Gauss-Seidel Method for Linear System (original) (raw)

A Modified Precondition in the Gauss-Seidel Method

Advances in Linear Algebra & Matrix Theory, 2012

In recent years, a number of preconditioners have been applied to solve the linear systems with Gauss-Seidel method (see [1-7,10-12,14-16]). In this paper we use S l instead of (S + S m) and compare with M. Morimoto's precondition [3] and H. Niki's precondition [5] to obtain better convergence rate. A numerical example is given which shows the preference of our method.

Comparison theorems of preconditioned Gauss–Seidel methods for M-matrices

Applied Mathematics and Computation, 2012

In this paper, a new preconditioner for the Gauss-Seidel method is proposed for solving linear systems whose coefficient matrix is an M-matrix. Several comparison theorems are shown for the proposed method with several preconditioners. It follows from the comparison results that our preconditioner is one of the best preconditioners in the sense of convergence rate. Finally, numerical examples are given to illustrate our theoretical results. Two conjectures are proposed as well based on our numerical tests.

Preconditioned Gauss-Seidel type iterative method for solving linear systems

Applied Mathematics and Mechanics, 2006

The preconditioned Gauss-Seidel type iterative method for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed. Also the optimal parameter is presented. Numerical results show that the proper choice of the preconditioner can lead to effective by the preconditioned Gauss-Seidel type iterative methods for solving linear systems.

A class of preconditioners based on the -type preconditioning matrices for solving linear systems

Applied Mathematics and Computation, 2007

The purpose of this paper is to present a class of preconditioners based on the ðI þ SðaÞÞ-type preconditioning matrices provided by Evans et al. [D.J. Evans, M.M. Martins, M.E. Trigo, The AOR iterative method for new preconditioned linear systems, J. Comput. Appl. Math. 132 (2001) 461-466] and Zhang et al. [Y. Zhang, T.Z. Huang, X.P. Liu, Modified iterative methods for nonnegative matrices and M-matrices linear systems, Comput. Math. Appl. 50 (2005) 1587-1602].

A collection of new preconditioners for solving linear systems

Scientific research and essays

In this paper, new preconditioners for solving linear systems are developed and preconditioned accelerated overrelaxation method (AOR) is used for the systems. The improvement of convergence rate via using new preconditioners method also shown. A numerical example is also given to illustrate our results. 2000 Mathematics Subject Classifications: 65F10, 15A06 Key Words and Phrases: linear systems, preconditioner, AOR iterative method, spectral radius, Z-, M- matrix

The upper Jacobi and upper Gauss–Seidel type iterative methods for preconditioned linear systems

Applied Mathematics Letters, 2006

The preconditioner for solving the linear system Ax = b introduced in [D.J. Evans, M.M. Martins, M.E. Trigo, The AOR iterative method for new preconditioned linear systems, J. Comput. Appl. Math. 132 -466] is generalized. Results obtained in this paper show that the convergence rate of Jacobi and Gauss-Seidel type methods can be increased by using the preconditioned method when A is an M-matrix.

A new model of (I+S)-type preconditioner for system of linear equations

2013

In this paper, we design a new model of preconditioner for systems of linear equations. The convergence properties of the proposed methods have been analyzed and compared with the classical methods. Numerical experiments of convection-diusi on equations show a good im- provement on the convergence, and show that the convergence rates of proposed methods are superior to the other modified iterative methods.

Convergence of Preconditioned Gauss-Seidel Iterative Method For −Matrices

Communication in Physical Sciences, 6(1): 803-808, 2020

A great many real-life situations are often modeled as linear system of equations, =. Direct methods of solution of such systems are not always realistic, especially where the coefficient matrix is very large and sparse, hence the recourse to iterative solution methods. The Gauss-Seidel, a basic iterative method for linear systems, is one such method. Although convergence is rarely guaranteed for all cases, it is established that the method converges for some situations depending on properties of the entries of the coefficient matrix and, by implication, on the algebraic structure of the method. However, as with all basic iterative methods, when it does converge, convergence could be slow. In this research, a preconditioned version of the Gauss-Seidel method is proposed in order to improve upon its convergence and robustness. For this purpose, convergence theorems are advanced and established. Numerical experiments are undertaken to validate results of the proved theorems.

A new preconditioner for indefinite and asymmetric matrices

Applied Mathematics and Computation, 2013

We present a novel preconditioner for numerical solutions of large sparse linear systems with indefinite and asymmetric matrices. This new preconditioner named as product preconditioner(PS) is constructed by two fairly simple preconditioners. The distribution of eigenvalues and the form of the eigenvectors of the preconditioned matrix are analyzed. Moreover, an upper bound on the degree of the minimal polynomial is also studied. Numerical experiments with several examples show that the proposed PS performs better than block diagonal preconditioner(BD) and block triangular preconditioner (BT) as well as the constraint preconditioner(SC) in terms of the number of iteration and computational time.

A preconditioning technique based on element matrix factorizations

Computer Methods in Applied Mechanics and Engineering, 1986

The task of making an incomplete factorization of the finite element stiffness matrix using only element matrices is concerned. We present a technique for realizing this and obtain a method which requires an amount of core storage that is independent of the number of unknowns in the discrete model, i.e., of the mesh size parameter. On the other hand datatransfers from/ to secondary storage and more arithmetic operations than in a corresponding completely-in-core method are required. For many problems solved in practice even thetotalrequirementofstorageforthestiffnessandpreconditioningmatricesisindependentofthesizeofthe mesh. Theoretical estimates of the rate of convergence of the corresponding preconditioned conjugate gradient method are derived for a model problem and a number of test examples are examined.