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1.
白中治  仇寿霞 《计算数学》2002,24(1):113-128
1.引 言 考虑大型稀疏线性代数方程组 为利用系数矩阵的稀疏结构以尽可能减少存储空间和计算开销,Krylov子空间迭代算法[1,16,23]及其预处理变型[6,8,13,18,19]通常是求解(1)的有效而实用的方法.当系数矩阵对称正定时,共轭梯度法(CG(  相似文献
2.
A QMR-based interior-point algorithm for solving linear programs   总被引:5,自引:0,他引:5  
A new approach for the implementation of interior-point methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2×2-block systems of linear equations that arise within the interior-point algorithm. These linear systems are solved by a symmetric variant of the quasi-minimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric 3×3-block systems to symmetric 2×2-block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interior-point algorithm is proposed. Some indefinite preconditioners are discussed. Finally, we report results of a few preliminary numerical experiments to illustrate the features of the new approach.  相似文献
3.
Lagrangian乘子区域分解法的一类预条件子   总被引:3,自引:2,他引:1       下载免费PDF全文
胡齐芽  梁国平 《计算数学》1998,20(2):201-212
1.引言非重叠区域分解的Lagrangian乘子法已被许多作者讨论[1今它是一类非协调区域分解法(与通常的非协调元区域分解不同),特别适合于非匹配网格的情形(即相邻子域在公共边或公共面上的结点不重合,参见14][6]).这种方法的一个最大优点是不要求界面变量在内交点(或内交边)上的连续性,从而界面方程易于建立,程序易于实现,而又正因为这个特点,使得界面矩阵的预条件子不能按通常的方法构造,故目前还未见到理想的预条件子(或者条件数差,或者应用上不方便).本文在很大程度上解决了这一问题.1)工作单位:湘潭大学数学系…  相似文献
4.
ON THE BREAKDOWNS OF THE GALERKIN AND LEAST-SQUARES METHODS   总被引:3,自引:0,他引:3  
1 IntroductionWeconsiderlinearsystemsoftheformAx=b,(1 )whereA∈CN×Nisnonsingularandpossiblynon Hermitian .Amajorclassofmethodsforsolving (1 )istheclassofKrylovsubspacemethods (see[6] ,[1 3]foroverviewsofsuchmethods) ,definedbythepropertiesxm ∈x0 +Km(r0 ,A) ;(2 )rm ⊥Lm, (3)whe…  相似文献
5.
The Davidson method is a preconditioned eigenvalue technique aimed at computing a few of the extreme (i.e., leftmost or rightmost) eigenpairs of large sparse symmetric matrices. This paper describes a software package which implements a deflated and variable-block version of the Davidson method. Information on how to use the software is provided. Guidelines for its upgrading or for its incorporation into existing packages are also included. Various experiments are performed on an SGI Power Challenge and comparisons with ARPACK are reported. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献
6.
For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.

  相似文献

7.
The purpose of this paper is to provide two numerical methods for solving the elastic body-plate problem by nonoverlapping domain decomposition type techniques, based on the discretization method by Wang. The first one is similar to an older method, but here the corresponding Schur complement matrix is preconditioned by a specific preconditioner associated with the plate problem. The second one is a ``displacement-force' type Schwarz alternating method. At each iteration step of the two methods, either a pure body or a pure plate problem needs to be solved. It is shown that both methods have a convergence rate independent of the size of the finite element mesh.

  相似文献

8.
SINE TRANSFORM MATRIX FOR SOLVING TOEPLITZ MATRIX PROBLEMS   总被引:2,自引:0,他引:2  
1. IntroductionStrang[1] first studied the use of circulallt matrices C for solving systems of linear eqllationsTi x = b witha symmetric positive definite Toeplitz matrix.Numerous authors such as T.Chan[2],R.Chan,etc.[3],[4],[5], Tyrtyshnikov[6], Huckle[7] and T.Ku and C.Kuo[8] proposed differentfamilies of circulallt / skew- circulant precondit ioners.Appling the preconditioned conjugate gradient algorithm(PCGA) to solve the systems Ti x -b, we must find a preconditioner P such that P…  相似文献
9.
SUBSTRUCTURE PRECONDITIONERS FOR NONCONFORMING PLATE ELEMENTS   总被引:2,自引:0,他引:2  
1.IntroductionInthispaper,wegeneralizetheBPSalgorithm[1]tononconformingelementfproximationsofthebiharmonicequation.WeconstructapreconditionerforMor:elementbysubstructuringonthebasisofafunctiondecompositionfordiscretebibmonicfunctions.Thefunctiondecomposit…  相似文献
10.
Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.  相似文献
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