A Modified Precondition in the Gauss-Seidel Method (original) (raw)

Effectiveness of Preconditioned m-Order Gauss-Seidel Method for Linear System

International Journal For Multidisciplinary Research

Focusing on the current and the proposed preconditioner, this work examines the efficacy of the preconditioned m-order Gauss-Seidel method. Type I + S and I+N preconditioning are used for the current and proposed preconditioner respectively. Preconditioning algorithms for a linear system are constructed using iterative approaches. MATLAB are used to get the findings. The effectiveness of iterative method is compared concerning convergence, condition number, determinant, spectral radius, and the number of iterations for the current and proposed preconditioner. The numerical results show that for a linear system, the preconditioned m-order Gauss-Seidel method converges at a faster rate and the proposed preconditioner succeeds where the current preconditioner fails.

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.

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.

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.

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 Review of Preconditioners for the Interval Gauss-Seidel Method

1991

Interval Newton methods in conjunction with generalized bi- section can form the basis of algorithms that Þnd all real roots within a speciÞed box X öRn of a system of nonlinear equations F(X) = 0 with mathematical certainty, even in Þnite-precision arithmetic. In such methods, the system F(X) = 0 is transformed into a linear interval system 0 = F(M)

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.

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

Generalized Jacobi and Gauss-Seidel Methods for Solving Linear System of Equations

2007

The Jacobi and Gauss-Seidel algorithms are among the stationary iterative methods for solving linear system of equations. They are now mostly used as preconditioners for the popular iterative solvers. In this paper a generalization of these methods are proposed and their convergence properties are studied. Some numerical experiments are given to show the efficiency of the new methods.

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].