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1.
Golub, Wu and Yuan [G.H. Golub, X. Wu, J.Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001) 71–85] have presented the SOR-like algorithm to solve augmented systems. In this paper, we present the modified symmetric successive overrelaxation (MSSOR) method for solving augmented systems, which is based on Darvishi and Hessari’s work above. We derive its convergence under suitable restrictions on the iteration parameter, determine its optimal iteration parameter and the corresponding optimal convergence factor under certain conditions. Finally, we apply the MSSOR method to solve augmented systems. 相似文献
2.
In this paper, we establish the generalized symmetric SOR method (GSSOR) for solving the large sparse augmented systems of linear equations, which is the extension of the SSOR iteration method. The convergence of the GSSOR method for augmented systems is studied. Numerical resume shows that this method is effective. 相似文献
3.
For the augmented system of linear equations, Golub, Wu and Yuan recently studied an SOR-like method (BIT 41(2001)71–85).
By further accelerating it with another parameter, in this paper we present a generalized SOR (GSOR) method for the augmented
linear system. We prove its convergence under suitable restrictions on the iteration parameters, and determine its optimal
iteration parameters and the corresponding optimal convergence factor. Theoretical analyses show that the GSOR method has
faster asymptotic convergence rate than the SOR-like method. Also numerical results show that the GSOR method is more effective
than the SOR-like method when they are applied to solve the augmented linear system. This GSOR method is further generalized
to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear
systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration. Besides, we also demonstrate
a complete version about the convergence theory of the SOR-like method.
Subsidized by The Special Funds For Major State Basic Research Projects (No. G1999032803) and The National Natural Science
Foundation (No. 10471146), P.R. China 相似文献
4.
In this paper, a new approach is proposed for solving the augmented systems. Based on the modified homotopy perturbation method, we construct the new iterative methods and derive the sufficient and necessary conditions for guaranteeing its convergence. Some numerical experiments show that this method is more simple and effective. 相似文献
5.
Li-Tao Zhang Ting-Zhu Huang Tong-Xiang Gu Xin-Lan Guo Jiang-Hua Yue 《Journal of Computational and Applied Mathematics》2009
Recently, Yun [Jae Heon Yun, Convergence of SSOR multisplitting method for an H-matrix, J. Comput. Appl. Math. 217 (2008) 252–258] studied the convergence of the relaxed multisplitting method associated with SSOR multisplitting for solving a linear system whose coefficient matrix is an H-matrix. In this paper, we improve the main results of Yun’s. Moreover, theoretical analysis and numerical examples clearly show that our new convergent domain is wider. 相似文献
6.
In this paper, the optimal iteration parameters of the symmetric successive overrelaxation (SSOR) method for a class of block two-by-two linear systems are obtained, which result in optimal convergence factor. An accelerated variant of the SSOR (ASSOR) method is presented, which significantly improves the convergence rate of the SSOR method. Furthermore, a more practical way to choose iteration parameters for the ASSOR method has also been proposed. Numerical experiments demonstrate the efficiency of the SSOR and ASSOR methods for solving a class of block two-by-two linear systems with the optimal parameters. 相似文献
7.
We apply Rouché's theorem to the functional equation relating the eigenvalues of theblock symmetric successive overrelaxation (SSOR) matrix with those of the block Jacobi iteration matrix found by Varga, Niethammer, and Cai, in order to obtain precise domains of convergence for the block SSOR iteration method associated withp-cyclic matricesA, p3. The intersection of these domains, taken over all suchp's, is shown to coincide with the exact domain of convergence of thepoint SSOR iteration method associated withH-matricesA. The latter domain was essentially discovered by Neumaier and Varga, but was recently sharpened by Hadjidimos and Neumann.Research supported in part by NSF Grant DMS 870064. 相似文献
8.
Recently, Lee et al. [Young-ju Lee, Jinbiao Wu, Jinchao Xu, Ludmil Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM J. Matrix Anal. Appl. 28 (2006) 634-641] introduce new criteria for the semi-convergence of general iterative methods for semidefinite linear systems based on matrix splitting. The new conditions generalize the classical notion of P-regularity introduced by Keller [H.B. Keller, On the solution of singular and semidefinite linear systems by iterations, SIAM J. Numer. Anal. 2 (1965) 281-290]. In view of their results, we consider here stipulations on a splitting A=M-N, which lead to fixed point systems such that, the iterative scheme converges to a weighted Moore-Penrose solution to the system Ax=b. Our results extend the result of Lee et al. to a more general case and we also show that it requires less restrictions on the splittings than Keller’s P-regularity condition to ensure the convergence of iterative scheme. 相似文献
9.
For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability
and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems,
namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite
versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite
preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning)
and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean
norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations
for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where
the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this
case.
This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative
Research Grant CRG 960782 相似文献
10.
In this paper, we consider a class of Uzawa-SOR methods for saddle point problems, and prove the convergence of the proposed methods. We solve a lower triangular system per iteration in the proposed methods, instead of solving a linear equation Az=b. Actually, the new methods can be considered as an inexact iteration method with the Uzawa as the outer iteration and the SOR as the inner iteration. Although the proposed methods cannot achieve the same convergence rate as the GSOR methods proposed by Bai et al. [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1-38], but our proposed methods have less workloads per iteration step. Experimental results show that our proposed methods are feasible and effective. 相似文献
11.
For solving 3D high order hierarchical FE systems the block SSOR preconditioned CG algorithms based on new stripwise block two-color orderings of degrees of freedom and providing for efficient concurrent/vector implementation are suggested. As demonstrated by numerical results for the 3D Navier equations approximated using hierarchical orderp, 2 p 5, FE's the convergence rate of such BSSOR-CG algorithms is only slightly dependent onp and mesh nonunformity. 相似文献
12.
We discuss the application of an augmented conjugate gradient to the solution of a sequence of linear systems of the same
matrix appearing in an iterative process for the solution of scattering problems. The conjugate gradient method applied to
the first system generates a Krylov subspace, then for the following systems, a modified conjugate gradient is applied using
orthogonal projections on this subspace to compute an initial guess and modified descent directions leading to a better convergence.
The scattering problem is treated via an Exact Controllability formulation and a preconditioned conjugate gradient algorithm
is introduced. The set of linear systems to be solved are associated to this preconditioning. The efficiency of the method
is tested on different 3D acoustic problems.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
13.
In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, we present block SSOR and modified block SSOR iteration methods based on the special structures of the coefficient matrices. In each step of the block SSOR iteration, we employ the block LU factorization to solve the sub-systems of linear equations. We show that the block LU factorization is existent and stable when the coefficient matrices are block diagonally dominant of type-II by columns. Under suitable conditions, we establish convergence theorems for both block SSOR and modified block SSOR iteration methods. In addition, the block SSOR iteration and AF-ADI method are considered as preconditioners for the nonsymmetric systems of linear equations. Numerical experiments show that both block SSOR and modified block SSOR iterations are feasible iterative solvers and they are also effective for preconditioning Krylov subspace methods such as GMRES and BiCGSTAB when used to solve this class of systems of linear equations. 相似文献
14.
Stefano Serra Capizzano 《Numerische Mathematik》2002,92(3):433-465
Summary. The solution of large Toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous
papers [13,14,22]. The technique was modified in [6,36] and a rigorous proof of convergence of the TGM (two-grid method) was
given in the special case where the generating function has only a zero at of order at most two. Here, by extending the latter approach, we perform a complete analysis of convergence of the TGM under
the sole assumption that f is nonnegative and with a zero at of finite order. An extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences
discretizing elliptic PDEs with nonconstant coefficients and of any order is then discussed.
Received May 28, 1999 / Revised version received January 26, 2001 / Published online November 15, 2001 相似文献
15.
A Newton method to solve total least squares problems for Toeplitz systems of equations is considered. When coupled with a
bisection scheme, which is based on an efficient algorithm for factoring Toeplitz matrices, global convergence can be guaranteed.
Circulant and approximate factorization preconditioners are proposed to speed convergence when a conjugate gradient method
is used to solve linear systems arising during the Newton iterations.
The work of the second author was partially supported by a National Science Foundation Postdoctoral Research Fellowship. 相似文献
16.
In this paper, some improvements on Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133] are presented for bounds of the spectral radius of lω,r, which is the iterative matrix of the generalized AOR (GAOR) method. Subsequently, some new sufficient conditions for convergence of GAOR method will be given, which improve some results of Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133]. 相似文献
17.
Summary.
Large, sparse nonsymmetric systems of linear equations with a
matrix whose eigenvalues lie in the right half plane may be solved by an
iterative method based on Chebyshev polynomials for an interval in the
complex plane. Knowledge of the convex hull of the spectrum of the
matrix is required in order to choose parameters upon which the
iteration depends. Adaptive Chebyshev algorithms, in which these
parameters are determined by using eigenvalue estimates computed by the
power method or modifications thereof, have been described by Manteuffel
[18]. This paper presents an adaptive Chebyshev iterative method, in
which eigenvalue estimates are computed from modified moments determined
during the iterations. The computation of eigenvalue estimates from
modified moments requires less computer storage than when eigenvalue
estimates are computed by a power method and yields faster convergence
for many problems.
Received May 13, 1992/Revised version received May 13,
1993 相似文献
18.
In this paper, we present some comparison theorems on preconditioned iterative method for solving Z-matrices linear systems, Comparison results show that the rate of convergence of the Gauss–Seidel-type method is faster than the rate of convergence of the SOR-type iterative method. 相似文献
19.
Zhong-Zhi Bai 《Annals of Operations Research》2001,103(1-4):263-282
A class of modified block SSOR preconditioners is presented for the symmetric positive definite systems of linear equations, whose coefficient matrices come from the hierarchical-basis finite-element discretizations of the second-order self-adjoint elliptic boundary value problems. These preconditioners include a block SSOR iteration preconditioner, and two inexact block SSOR iteration preconditioners whose diagonal matrices except for the (1,1)-block are approximated by either point symmetric Gauss–Seidel iterations or incomplete Cholesky factorizations, respectively. The optimal relaxation factors involved in these preconditioners and the corresponding optimal condition numbers are estimated in details through two different approaches used by Bank, Dupont and Yserentant (Numer. Math. 52 (1988) 427–458) and Axelsson (Iterative Solution Methods (Cambridge University Press, 1994)). Theoretical analyses show that these modified block SSOR preconditioners are very robust, have nearly optimal convergence rates, and especially, are well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes. 相似文献
20.
Yiannis G. Saridakis 《BIT Numerical Mathematics》1986,26(3):369-376
In this paper, the behavior of the block Accelerated Overrelaxation (AOR) iterative method, when applied to linear systems with a generalized consistently ordered coefficient matrix, is investigated. An equation, relating the eigenvalues of the block Jacobi iteration matrix to the eigenvalues of its associated block AOR iteration matrix, as well as sufficient conditions for the convergence of the block AOR method, are obtained. 相似文献