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1.
Recently Y. Saad proposed a flexible inner-outer preconditioned GMRES algorithm for nonsymmetric linear systems [4]. Following their ideas, we suggest an adaptive preconditioned CGS method, called CGS/GMRES (k), in which the preconditioner is constructed in the iteration step of CGS, by several steps of GMRES(k). Numerical experiments show that the residual of the outer iteration decreases rapidly. We also found the interesting residual behaviour of GMRES for the skewsymmetric linear system Ax = b, which gives a convergence result for restarted GMRES (k). For convenience, we discuss real systems. 相似文献
2.
It is well-known that Bi-CG can be adapted so that the operations withA
T can be avoided, and hybrid methods can be constructed in which it is attempted to further improve the convergence behaviour. Examples of this are CGS, Bi-CGSTAB, and the more general BiCGstab(l) method. In this paper it is shown that BiCGstab(l) can be implemented in different ways. Each of the suggested approaches has its own advantages and disadvantages. Our implementations allow for combinations of Bi-CG with arbitrary polynomial methods. The choice for a specific implementation can also be made for reasons of numerical stability. This aspect receives much attention. Various effects have been illustrated by numerical examples. 相似文献
3.
This paper gives the truncated version of the Minpert method:the incomplete minimum perturbation algorithm(IMinpert).It is based on an incomplete orthogonal- ization of the Krylov vectors in question,and gives a quasi-minimum backward error solution over the Krylov subspace.In order to make the practical implementation of IMinpert easy and convenient,we give another approximate version of the IMinpert method:A-IMinpert.Theoretical properties of the latter algorithm are discussed.Nu- merical experiments are reported to show the proposed method is effective in practice and is competitive with the Minpert algorithm. 相似文献
4.
王正盛 《高等学校计算数学学报(英文版)》2001,10(2)
1 IntroductionThe solution of large N× N nonsingular unsymmetric( non-Hermitian) sparse sys-tems of linear equationsAx =b, ( 1 )is one of the most frequently encountered tasks in numerical computations.For example,such systems arise from finite difference or finite element approximations to partial differ-ential equationsA major class of methods for solving ( 1 ) is Krylov subspace or conjugate gradienttype methods.Most successful scheme of these methods are based on the orthogonal pro-jec… 相似文献
5.
Iterative methods and especially Krylov subspace methods (KSM) are a very useful numerical tool in solving for large and sparse linear systems problems arising in science and engineering modeling. More recently, the nested loop KSM have been proposed that improve the convergence of the traditional KSM. In this article, we review the residual cutting (RC) and the generalized residual cutting (GRC) that are nested loop methods for large and sparse linear systems problems. We also show that GRC is a KSM that is equivalent to Orthomin with a variable preconditioning. We use the modified Gram–Schmidt method to derive a stable GRC algorithm. We show that GRC presents a general framework for constructing a class of “hybrid” (nested) KSM based on inner loop method selection. We conduct numerical experiments using nonsymmetric indefinite matrices from a widely used library of sparse matrices that validate the efficiency and the robustness of the proposed methods. 相似文献
6.
Lanczos‐type product methods (LTPMs), in which the residuals are defined by the product of stabilizing polynomials and the Bi‐CG residuals, are effective iterative solvers for large sparse nonsymmetric linear systems. Bi‐CGstab(L) and GPBi‐CG are popular LTPMs and can be viewed as two different generalizations of other typical methods, such as CGS, Bi‐CGSTAB, and Bi‐CGStab2. Bi‐CGstab(L) uses stabilizing polynomials of degree L, while GPBi‐CG uses polynomials given by a three‐term recurrence (or equivalently, a coupled two‐term recurrence) modeled after the Lanczos residual polynomials. Therefore, Bi‐CGstab(L) and GPBi‐CG have different aspects of generalization as a framework of LTPMs. In the present paper, we propose novel stabilizing polynomials, which combine the above two types of polynomials. The resulting method is referred to as GPBi‐CGstab(L). Numerical experiments demonstrate that our presented method is more effective than conventional LTPMs. 相似文献
7.
Flexible and multi‐shift induced dimension reduction algorithms for solving large sparse linear systems 下载免费PDF全文
Martin B. van Gijzen Gerard L. G. Sleijpen Jens‐Peter M. Zemke 《Numerical Linear Algebra with Applications》2015,22(1):1-25
We give two generalizations of the induced dimension reduction (IDR) approach for the solution of linear systems. We derive a flexible and a multi‐shift quasi‐minimal residual IDR variant. These variants are based on a generalized Hessenberg decomposition. We present a new, more stable way to compute basis vectors in IDR. Numerical examples are presented to show the effectiveness of these new IDR variants and the new basis compared with existing ones and to other Krylov subspace methods. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
8.
In this paper, a generalized global conjugate gradient squared method for solving nonsymmetric linear systems with multiple right-hand sides is presented. The method can be derived by using products of two nearby global BiCG polynomials and formal orthogonal polynomials, of which global CGS and global BiCGSTAB are just particular cases. We also show to apply the method for solving the Sylvester matrix equation. Finally, numerical examples are given to illustrate the effectiveness of the proposed method. 相似文献
9.
10.
ON THE BREAKDOWNS OF THE GALERKIN AND LEAST-SQUARES METHODS 总被引:3,自引:0,他引:3
钟宝江 《高等学校计算数学学报(英文版)》2002,11(2):137-148
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
11.
Tatiana S. Martynova 《计算数学(英文版)》2014,32(3):297-305
An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitting iteration methods. We consider the saddle-point linear systems with singular or semidefinite (1, 1) blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse saddle-point linear systems. 相似文献
12.
Kuniyoshi Abe Shao-Liang Zhang Taketomo Mitsui Cheng-Hai Jin 《Numerical Algorithms》2004,36(3):189-202
For singular linear systems A
x=b, ORTHOMIN(2) is known theoretically to attain the minimum residual min
xR
nb–A
x2 under a certain condition. However, in the actual computation with finite precision arithmetic, the residual is often observed to be reduced further than the theoretically expected level. Therefore, we propose a variant of ORTHOMIN(2), which is mathematically equivalent to the original ORTHOMIN(2) method, but uses recurrence formulas that are different from those of ORTHOMIN(2); they contain alternative expressions for the auxiliary vector and the recurrence coefficients. Although our implementation has the same computational costs as ORTHOMIN(2), numerical experiments on singular systems show that our implementation is more accurate and less affected by rounding errors than ORTHOMIN(2). 相似文献
13.
The block‐Lanczos method serves to compute a moderate number of eigenvalues and the corresponding invariant subspace of a symmetric matrix. In this paper, the convergence behavior of nonrestarted and restarted versions of the block‐Lanczos method is analyzed. For the nonrestarted version, we improve an estimate by Saad by means of a change of the auxiliary vector so that the new estimate is much more accurate in the case of clustered or multiple eigenvalues. For the restarted version, an estimate by Knyazev is generalized by extending our previous results on block steepest descent iterations and single‐vector restarted Krylov subspace iterations. The new estimates can also be reformulated and applied to invert‐block‐Lanczos methods for solving generalized matrix eigenvalue problems. 相似文献
14.
Tijmen P. Collignon Martin B. van Gijzen 《Numerical Linear Algebra with Applications》2011,18(5):805-825
IDR (s) is a family of fast algorithms for iteratively solving large nonsymmetric linear systems. With cluster computing and in particular with Grid computing, the inner product is a bottleneck operation. In this paper, three techniques are investigated for alleviating this bottleneck. First, a recently proposed IDR (s) algorithm that is highly efficient and stable is reformulated in such a way that it has a single global synchronization point per iteration step. Second, the so‐called test matrix is chosen so that the work, communication, and storage involving this matrix is minimized in multi‐cluster environments. Finally, a methodology is presented for a‐priori estimation of the optimal value of s using only problem and machine‐based parameters. Numerical experiments applied to a 3D convection–diffusion problem are performed on the DAS‐3 Grid computer, demonstrating the effectiveness of our approach. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
15.
Inexact Newton methods are variant of the Newton method in which each step satisfies only approximately the linear system (Ref. 1). The local convergence theory given by the authors of Ref. 1 and most of the results based on it consider the error terms as being provided only by the fact that the linear systems are not solved exactly. The few existing results for the general case (when some perturbed linear systems are considered, which in turn are not solved exactly) do not offer explicit formulas in terms of the perturbations and residuals. We extend this local convergence theory to the general case, characterizing the rate of convergence in terms of the perturbations and residuals.The Newton iterations are then analyzed when, at each step, an approximate solution of the linear system is determined by the following Krylov solvers based on backward error minimization properties: GMRES, GMBACK, MINPERT. We obtain results concerning the following topics: monotone properties of the errors in these Newton–Krylov iterates when the initial guess is taken 0 in the Krylov algorithms; control of the convergence orders of the Newton–Krylov iterations by the magnitude of the backward errors of the approximate steps; similarities of the asymptotical behavior of GMRES and MINPERT when used in a converging Newton method. At the end of the paper, the theoretical results are verified on some numerical examples. 相似文献
16.
We present, implement and test several incomplete QR factorization methods based on Givens rotations for sparse square and rectangular matrices. For square systems, the approximate QR factors are used as right-preconditioners for GMRES, and their performance is compared to standard ILU techniques. For rectangular matrices corresponding to linear least-squares problems, the approximate R factor is used as a right-preconditioner for CGLS. A comprehensive discussion is given about the uses, advantages and shortcomings of the preconditioners.
AMS subject classification (2000) 65F10, 65F25, 65F50.Received May 2002. Revised October 2004. Communicated by Åke Björck. 相似文献
17.
A note on block preconditioner for generalized saddle point matrices with highly singular (1,1) block 下载免费PDF全文
Litao Zhang Yongwei Zhou Xianyu Zuo Chaoqian Li Yaotang Li 《Journal of Applied Analysis & Computation》2019,9(3):916-927
In this paper, we present a block triangular preconditioner for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Theoretical analysis shows that all the eigenvalues of the preconditioned matrix are strongly clustered when choosing an optimal parameter. Numerical experiments are given to demonstrate the efficiency of the presented preconditioner. 相似文献
18.
Consecutive-(r,f,k)-out-of-n:F系统由n个单元顺序连结而成,仅当在连续的r个单元中,至少有f个失效或者至少连续k个失效,整个系统才失效;而Consecutive-(f,g)-out-of-(r,n):F系统由n个单元顺序连结而成,仅当在整个系统中至少有f个失效或者在连续的r个单元中,至少有g个失效,整个系统才失效。本文运用马氏链嵌入方法,在单元之间相互独立以及单元之间马氏相关这两种情况下,给出线性系统的可靠性。 相似文献
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20.
P. Amodio J. R. Cash G. Roussos R. W. Wright G. Fairweather I. Gladwell G. L. Kraut M. Paprzycki 《Numerical Linear Algebra with Applications》2000,7(5):275-317
Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two‐point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently. We survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise. We outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, we examine parallel iterative methods for solving BABD systems. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献