• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1463-1475
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• 2011

This paper proposes a new generalized iterative method (SSOR-like method) for solving augmented system. A functional equation relating two involved parameters is obtained, and some convergence conditions for this method are derived. This paper generalizes some foregone results. Numerical examples show that, this method is efficient by suitable choices of the involved parameters.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.657-670
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• 2019

In this paper, we consider the preconditioned iterative methods for solving linear complementarity problem associated with an M-matrix. Based on the generalized Gunawardena's preconditioner, two preconditioned SSOR methods for solving the linear complementarity problem are proposed. The convergence of the proposed methods are analyzed, and the comparison results are derived. The comparison results showed that preconditioned SSOR methods accelerate the convergent rate of the original SSOR method. Numerical examples are used to illustrate the theoretical results.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.273-282
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• 2007

In this paper, we study the convergence of both the multisplitting method and the relaxed multisplitting method associated with SOR or SSOR multisplittings for solving a linear system whose coefficient matrix is an M-matrix.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.305-316
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• 2003

Recently, an iterative algorithm for finding the interior eigenvalues of a definite matrix by CG-type method has been proposed. This method compares to the inverse power method. The given matrices A, and B are assumed to be large and sparse, and SPD( Symmetric Positive Definite) The CG scheme for the optimization of the Rayleigh quotient has been proven a very attractive and promising technique for large sparse eigenproblems for smallest eigenvalue. Also, it is very amenable to parallel computations, like the CG method for the linear systems. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. But for parallel computations we need to find an efficient parallel preconditioner. Our candidates we ILU(0) in the wave-front order, ILU(0) in the multi-coloring order, Point-SSOR(Symmetric Successive Overrelaxation), and Multi-Color Block SSOR preconditioner. Wavefront order is a simple way to increase parallelism in the natural order, and Multi-coloring realizes a parallelism of order(N), where N is the order of the matrix. Another choice is the Multi-Color Block SSOR(Symmetric Successive OverRelaxation) preconditioning. Block SSOR is a symmetric preconditioner which is expected to minimize the interprocessor communication due to the blocking. We implemented the results on the CRAY-T3E with 128 nodes. The MPI (Message Passing Interface) library was adopted for the interprocessor communications. The test problem was drawn from the discretizations of partial differential equations by finite difference methods. The results show that for small number of processors Multi-Color ILU(0) has the best performance, while for large number of processors Multi-Color Block SSOR performs the best.

• The KIPS Transactions:PartA
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• v.10A no.2
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• pp.137-140
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• 2003

Recently, CG (Conjugate Gradient) scheme for the optimization of the Rayleigh quotient has been proven a very attractive and promising technique for interior eigenvalues for the following eigenvalue problem, Ax＝λx (1) The given matrix A is assummed to be large and sparse, and symmetric. Also, the method is very amenable to parallel computations. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. We compare the parallel preconditioners for the computation of the interior eigenvalues of a symmetric matrix by CG-type method. The considered preconditioners are Point-SSOR, ILU (0) in the multi-coloring order, and Multi-Color Block SSOR (Symmetric Succesive OverRelaxation). We conducted our experiments on the CRAY­T3E with 128 nodes. The MPI (Message Passing Interface) library was adopted for the interprocessor communications. The test matrices are up to $512{\times}512$ in dimensions and were created from the discretizations of the elliptic PDE. All things considered the MC-BSSOR seems to be most robust preconditioner.

• Journal of the Chungcheong Mathematical Society
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• v.10 no.1
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• pp.57-70
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• 1997

In this paper two preconditioned GMRES and QMR methods are applied to the non-Hermitian system from the Petrov-Galerkin procedure for the Poisson equation and compared to each other. To our purpose the ILUT and the SSOR preconditioners are used.

• Journal of applied mathematics & informatics
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• v.30 no.3_4
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• pp.593-602
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• 2012

In this paper, we first introduce a special type of multisplitting method with different weighting scheme, and then we provide convergence results of multisplitting methods with different weighting schemes corresponding to both the AOR-like multisplitting and the SSOR-like multisplitting.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.997-1006
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• 2012

In this paper, we study convergence of multisplitting methods with preweighting for solving a linear system whose coefficient matrix is an H-matrix corresponding to both the AOR multisplitting and the SSOR multisplitting. Numerical results are also provided to confirm theoretical results for the convergence of multisplitting methods with preweighting.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.5 no.2
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• pp.63-69
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• 2001

In this study, we shall be concerned with computing in parallel a few of the smallest eigenvalues and their corresponding eigenvectors of the eigenvalue problem, $Ax={\lambda}Bx$, where A is symmetric, and B is symmetric positive definite. Both A and B are large and sparse. Recently iterative algorithms based on the optimization of the Rayleigh quotient have been developed, and CG scheme for the optimization of the Rayleigh quotient has been proven a very attractive and promising technique for large sparse eigenproblems for small extreme eigenvalues. As in the case of a system of linear equations, successful application of the CG scheme to eigenproblems depends also upon the preconditioning techniques. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. The idea underlying the present work is a parallel computation of the Multi-Color Block SSOR preconditioning for the CG optimization of the Rayleigh quotient together with deflation techniques. Multi-Coloring is a simple technique to obatin the parallelism of order n, where n is the dimension of the matrix. Block SSOR is a symmetric preconditioner which is expected to minimize the interprocessor communication due to the blocking. We implemented the results on the CRAY-T3E with 128 nodes. The MPI(Message Passing Interface) library was adopted for the interprocessor communications. The test problems were drawn from the discretizations of partial differential equations by finite difference methods.

• The KIPS Transactions:PartA
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• v.8A no.3
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• pp.227-234
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• 2001

In this paper we propose a block-type parallel preconditioner for solving large sparse nonsymmetric linear systems, which we expect to be scalable. It is Multi-Color Block SOR preconditioner, combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a nondeteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. We compared it with four other preconditioners, which are ILU(0)-wavefront ordering, ILU(0)-Multi-Color ordering, SPAI(SParse Approximate Inverse), and SSOR preconditiner. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to $1025{\times}1024$. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications, The results show that Multi-Color Block SOR is scalabl and gives the best performances.