A modified SSOR iterative method for augmented systems

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.     

Application of modified homotopy perturbation method for solving the augmented systems

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.     

Preconditioned Richardson iteration for augmented linear systems

Numerical Algorithms - For solving a class of augmented linear systems, we propose a new efficient iteration method, which is called preconditioned Richardson iteration (PR). Under suitable...     

A modified predictor-corrector method for linear programming

In the predictor-corrector method of Mizuno, Todd and Ye [1], the duality gap is reduced only at the predictor step and is kept unchanged during the corrector step. In this paper, we modify the corrector step so that the duality gap is reduced by a constant fraction, while the predictor step remains unchanged. It is shown that this modified predictor-corrector method retains the iteration complexity as well as the local quadratic convergence property.     

A note on an SOR-like method for augmented systems

Golub et al. (2001, BIT, 41, 71–85) gave a generalizedsuccessive over-relaxation method for the augmented systems.In this paper, the connection between the SOR-like method andthe preconditioned conjugate gradient (PCG) method for the augmentedsystems is investigated. It is shown that the PCG method isat least as accurate (fast) as the SOR-like method. Numericalexamples demonstrate that the PCG method is much faster thanthe SOR-like method.     

Imposing symmetry in augmented linear systems

This paper discusses the methods of imposing symmetry in the augmented system formulation (ASF) for least‐squares (LS) problems. A particular emphasis is on upper Hessenberg problems, where the challenge lies in leaving all zero‐by‐definition elements of the LS matrix unperturbed. Analytical solutions for optimal perturbation matrices are given, including upper Hessenberg matrices. Finally, the upper Hessenberg LS problems represented by unsymmetric ASF that indicate a normwise backward stability of the problem (which is not the case in general) are identified. It is observed that such problems normally arise from Arnoldi factorization (for example, in the generalized minimal residual (GMRES) algorithm). The problem is illustrated with a number of practical (arising in the GMRES algorithm) and some ‘purpose‐built’ examples. Copyright © 2014 John Wiley & Sons, Ltd.     

DGMRES method augmented with eigenvectors for computing the Drazin-inverse solution of singular linear systems

Acta Mathematicae Applicatae Sinica, English Series - The DGMRES method for solving Drazin-inverse solution of singular linear systems is generally used with restarting. But the restarting often...     

An augmented LDG method for linear diffusion problems

In this note we present a review of a stabilized discontinuous Galerkin method for elliptic problems in the plane with mixed boundary conditions. The stabilized scheme is obtained by adding suitable Galerkin least-squares terms. The corresponding unique solvability and optimal rates of convergence, with respect to the h –version, are established by applying the wellknown Lax-Milgram theorem, avoiding therefore the introduction of any lifting operator for the analysis. Furthermore, we include a reliable and efficient (up to high order terms) a posteriori error estimator. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On generalized successive overrelaxation methods for augmented linear systems

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     

A modified damped Newton method for linear complementarity problems

We present a modified damped Newton method for solving large sparse linear complementarity problems, which adopts a new strategy for determining the stepsize at each Newton iteration. The global convergence of the new method is proved when the system matrix is a nondegenerate matrix. We then apply the matrix splitting technique to this new method, deriving an inexact splitting method for the linear complementarity problems. The global convergence of the resulting inexact splitting method is proved, too. Numerical results show that the new methods are feasible and effective for solving the large sparse linear complementarity problems.     

On generalized symmetric SOR method for augmented systems

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.     

Solution of augmented linear systems using orthogonal factorizations

We consider solvingx+Ay=b andA ^T x=c for givenb, c andm ×n A of rankn. This is called the augmented system formulation (ASF) of two standard optimization problems, which include as special cases the minimum 2-norm of a linear underdetermined system (b=0) and the linear least squares problem (c=0), as well as more general problems. We examine the numerical stability of methods (for the ASF) based on the QR factorization ofA, whether by Householder transformations, Givens rotations, or the modified Gram-Schmidt (MGS) algorithm, and consider methods which useQ andR, or onlyR. We discuss the meaning of stability of algorithms for the ASF in terms of stability of algorithms for the underlying optimization problems.We prove the backward stability of several methods for the ASF which useQ andR, inclusing a new one based on MGS, and also show under what circumstances they may be regarded as strongly stable. We show why previous methods usingQ from MGS were not backward stable, but illustrate that some of these methods may be acceptable-error stable. We point out that the numerical accuracy of methods that do not useQ does not depend to any significant extent on which of of the above three QR factorizations is used. We then show that the standard methods which do not useQ are not backward stable or even acceptable-error stable for the general ASF problem, and discuss how iterative refinement can be used to counteract these deficiencies.Dedicated to Carl-Eric Fröberg on the occasion of his 75th birthdayThis research was partially supported by NSERC of Canada Grant No. A9236.     

A new augmented Lyapunov-Krasovskii functional approach for stability of linear systems with time-varying delays

This paper proposes improved delay-dependent conditions for asymptotic stability of linear systems with time-varying delays. The proposed method employs a suitable Lyapunov-Krasovskii’s functional for new augmented system. Based on Lyapunov method, delay-dependent stability criteria for the systems are established in terms of linear matrix inequalities (LMIs) which can be easily solved by various optimization algorithms. Three numerical examples are included to show that the proposed method is effective and can provide less conservative results.     

A discretization method for systems of linear inequalities

We investigate a method for approximating a convex domainCR ⁿ described by a (possibly infinite) set of linear inequalities. In contrast to other methods, the approximating domains (polyhedrons) lie insideC. We discuss applications to semi-infinite programming and present numerical examples.The paper was written at the Institut für Angewandte Mathematik, Universität Hamburg, Hamburg, West Germany. The author thanks Prof. U. Eckhardt, Dr. K. Roleff, and Prof. B. Werner for helpful discussions.     

Convergence of a generalized MSSOR method for augmented systems

Recently, Wu et al. [S.-L. Wu, T.-Z. Huang, X.-L. Zhao, A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math. 228 (1) (2009) 424-433] introduced a modified SSOR (MSSOR) method for augmented systems. In this paper, we establish a generalized MSSOR (GMSSOR) method for solving the large sparse augmented systems of linear equations, which is the extension of the MSSOR method. Furthermore, the convergence of the GMSSOR method for augmented systems is analyzed and numerical experiments are carried out, which show that the GMSSOR method with appropriate parameters has a faster convergence rate than the MSSOR method with optimal parameters.     

A COCR method for solving complex symmetric linear systems

The Conjugate Orthogonal Conjugate Gradient (COCG) method has been recognized as an attractive Lanczos-type Krylov subspace method for solving complex symmetric linear systems; however, it sometimes shows irregular convergence behavior in practical applications. In the present paper, we propose a Conjugate A$A$-Orthogonal Conjugate Residual (COCR) method, which can be regarded as an extension of the Conjugate Residual (CR) method. Numerical examples show that COCR often gives smoother convergence behavior than COCG.     

A hybrid iterative method for symmetric indefinite linear systems

Hybrid iterative methods that combine a conjugate direction method with a simpler iteration scheme, such as Chebyshev or Richardson iteration, were first proposed in the 1950s. The ease with which Chebyshev and Richardson iteration can be implemented efficiently on a large variety of computer architectures has in recent years lead to renewed interest in iterative methods that use Chebyshev or Richardson iteration. This paper presents a new hybrid iterative method for the solution of linear systems of equations with a symmetric indefinite matrix. Our method combines the conjugate residual method with Richardson iteration. Special attention is paid to the determination of two real intervals, one on each side of the origin, that contain most of the eigenvalues of the matrix. These intervals are used to compute suitable iteration parameters for Richardson iteration. We also discuss when to switch between the methods. The hybrid scheme typically uses the Richardson method for most iterations, and this reduces the number of arithmetic vector operations significantly compared with the number of arithmetic vector operations required when only the conjugate residual method is used. Computed examples illustrate the competitiveness of the hybrid scheme.     

A stable Richardson iteration method for complex linear systems

Summary The Richardson iteration method is conceptually simple, as well as easy to program and parallelize. This makes the method attractive for the solution of large linear systems of algebraic equations with matrices with complex eigenvalues. We change the ordering of the relaxation parameters of a Richardson iteration method proposed by Eiermann, Niethammer and Varga for the solution of such problems. The new method obtained is shown to be stable and to have better convergence properties.Research supported by the National Science Foundation under Grant DMS-8704196