Iterative Methods For Linear Systems Research Articles

Modified alternating direction-implicit iteration method for linear systems from the incompressible Navier–Stokes equations

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, Ran and Yuan [On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems, Appl. Math. Comput. 217 (2010), pp. 3050–3068] presented the block symmetric successive over-relaxation (BSSOR) and the modified BSSOR iteration methods based on the special structures of the coefficient matrices. In this study, we present the modified alternating direction-implicit (MADI) iteration method for solving the linear systems. Under suitable conditions, we establish convergence theorems for the MADI iteration method. In addition, the optimal parameter involved in the MADI iteration method is estimated in detail. Numerical experiments show that the MADI iteration method is a feasible and effective iterative solver.

On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems

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.

Convergence Analysis of Preconditioned AOR Iterative Method for Linear Systems

M-(H-)matrices appear in many areas of science and engineering, for example, in the solution of the linear complementarity problem (LCP) in optimization theory and in the solution of large systems for real-time changes of data in fluid analysis in car industry. Classical (stationary) iterative methods used for the solution of linear systems have been shown to convergence for this class of matrices. In this paper, we present some comparison theorems on the preconditioned AOR iterative method for solving the linear system. Comparison results show that the rate of convergence of the preconditioned iterative method is faster than the rate of convergence of the classical iterative method. Meanwhile, we apply the preconditioner toH-matrices and obtain the convergence result. Numerical examples are given to illustrate our results.

Thermal aware floorplanning using Gauss-Seidel method

The Gauss-Seidel method is effective to solve the traditional sparse linear system. In the paper, we define a class of sparse linear systems in iterative algorithm. The iterative method for linear system can be extended to the dummy sparse linear system. We apply the Gauss-Seidel method, which is one of the iterative methods for linear system, to the thermal model of floorplan of VLSI physical design. The experimental results of dummy sparse linear system are computed by using Gauss-Seidel method that have shown our theory analysis and extendibility. The iterative time of our incremental thermal model is 5 times faster than that of the inverting matrix method.

Convergence of block iterative methods for linear systems with generalized H-matrices

The paper studies the convergence of some block iterative methods for the solution of linear systems when the coefficient matrices are generalized H -matrices. A truth is found that the class of conjugate generalized H -matrices is a subclass of the class of generalized H -matrices and the convergence results of R. Nabben [R. Nabben, On a class of matrices which arises in the numerical solution of Euler equations, Numer. Math. 63 (1992) 411–431] are then extended to the class of generalized H -matrices. Furthermore, the convergence of the block AOR iterative method for linear systems with generalized H -matrices is established and some properties of special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper.

A multi-parameters preconditioned AOR iterative method for linear systems

In order to improve the convergence rates of iterative method solving the linear system Ax = b , a preconditioned AOR iterative method with a multi-parameters preconditioner I + S ∼ α is proposed. Some convergence and comparison results for α i ∈ [ 0 , 1 ] are given when A is an L-matrix. When A is an H-matrix, the convergence of the corresponding methods is discussed for α i ∈ [ 0 , α i ′ ) , where α i > 1 . Furthermore, effectiveness of the new preconditioner method is shown by numerical experiments. Numerical experiments show that our methods are superior to the basic AOR iterative methods.

Discrete mathematical models in the analysis of splitting iterative methods for linear systems

Splitting methods are used to solve most of the linear systems, A x = b , when the conventional method of Gauss is not efficient. These methods use the factorization of the square matrix A into two matrices M and N as A = M − N where M is nonsingular. Basic iterative methods such as Jacobi or Gauss–Seidel define the iterative scheme for matrices that have no zeros along its main diagonal. This paper is concerned with the development of an iterative method to approximate solutions when the coefficient matrix A has some zero diagonal entries. The algorithm developed in this paper involves the analysis of a discrete-time descriptor system given by the equation M e ( k + 1 ) = N e ( k ) , e ( k ) being the error vector.

Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices

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 l ω , 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].

Convergence theorems for block splitting iterative methods for linear systems

In this paper, we will present the block splitting iterative methods with general weighting matrices for solving linear systems of algebraic equations Ax = b when the coefficient matrix A is symmetric positive definite of block form, and establish the convergence theories with respect to the general weighting matrices but special splittings. Finally, a numerical example shows the advantage of this method.

Preconditioned AOR iterative method for linear systems

For solving the linear system A x = b , different preconditioned Gauss–Seidel methods have been proposed by many authors. In this paper, we will present preconditioned AOR iterative methods with two different preconditioners, and give corresponding convergence and comparison results. Numerical example is also given to illustrate our results.

Iterative methods for linear systems: following the meandering way

Solving a linear system of equations is one of the easiest computational tasks imaginable, but when the matrix is very large, the algorithms we learned in linear algebra class take too long, and we need to use a different type of method. In this homework assignment, we investigate these iterative methods.

Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices

The optimal parameter of the Hermitian/skew-Hermitian splitting (HSS) iteration method for a real two-by-two linear system is obtained. The result is used to determine the optimal parameters for linear systems associated with certain two-by-two block matrices and to estimate the optimal parameters of the HSS iteration method for linear systems with n-by-n real coefficient matrices. Numerical examples are given to illustrate the results.

Convergence analysis of Krylov subspace methods

AbstractOne of the most powerful tools for solving large and sparse systems of linear algebraic equations is a class of iterative methods called Krylov subspace methods. Their significant advantages like low memory requirements and good approximation properties make them very popular, and they are widely used in applications throughout science and engineering. The use of the Krylov subspaces in iterative methods for linear systems is even counted among the “Top 10” algorithmic ideas of the 20th century. Convergence analysis of these methods is not only of a great theoretical importance but it can also help to answer practically relevant questions about improving the performance of these methods. As we show, the question about the convergence behavior leads to complicated nonlinear problems. Despite intense research efforts, these problems are not well understood in some cases. The goal of this survey is to summarize known convergence results for three well‐known Krylov subspace methods (CG, MINRES and GMRES) and to formulate open questions in this area. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Matrix Algorithms, Volume II: Eigensystems

1R3. Matrix Algorithms, Volume II: Eigensystems. - GW Stewart (Dept of Comput Sci, Univ of Maryland, College Park MD). SIAM, Philadelphia. 2001. 469 pp. (Softcover). ISBN 0-89871-503-2. $50.00. Reviewed by A Mahajan (Dept of Mech Eng and Energy Processes, S Illinois Univ, Carbondale IL 62901).This book contains a comprehensive presentation of computations involving the eigenvalues and eigenvectors of a matrix. It is the second volume in a projected five-volume series on matrix algorithms. The first volume was about basic decompositions. The second volume (this book) presents eigensystems. The next three books are projected to treat iterative methods for linear systems, sparse direct methods, and special topics, including fast algorithms for structured matrices. The author has managed to keep this volume fairly independent of the first volume, hence a basic knowledge of linear algebra is sufficient to understand most topics in the book is necessary. The book is really intended to be a reference book for scientists and engineers who need tools to solve problems in eigensystems. According to the author, the intended audience is the nonspecialist whose needs cannot be satisfied by black boxes. The book does live up to the author’s stated aim. The author gives detailed derivations that will help anyone who wants to adapt the methods to particular problems. The book really is not intended to be a textbook due to the lack of extensive solved and unsolved problems, but can be used as a self study by graduate students or even undergraduate students in honors programs. The book is divided into two parts: Dense Eigenproblems and Large Eigenproblems. The book has six chapters, a reference list of over 300 references, and an excellent subject index. The first chapter presents the underlying theory for eigensystems. The second chapter describes the widely used QR algorithm. Chapter 3 deals with symmetric matrices and the singular value decomposition. The fourth chapter deals with large matrices for which the computation of a complete eigensystem is not possible, hence an algebraic and analytic theory of eigensystems is presented. The fifth chapter deals with the Krylov sequence methods, in particular the Arnoldi and Lanczos methods. The sixth chapter presents alternative methods such as subspace iteration and the Jacobi-Davidson method. The book provides pseudo codes for many algorithms. The author does admit that the pseudo codes are for illustration only and should not be regarded as finished implementations, and it is difficult to verify their correctness. He also mentions that whereever possible he has checked the algorithms against MATLAB implementations. If there is a weakness in the book, it is right here, for the author could have provided these MATLAB programs on disk, or at the very least, made them available on a website. The reviewer did implement some of the algorithms in MATLAB, and it did take some time to develop the implementation. Of course, this in no way impacts the significance of the material to the serious reader who is looking for a detailed treatise on the subject, but does impact a student or practicing engineer who could be looking for a quick implementation of a particular algorithm. A recommendation to the author and publisher would be to make these MATLAB programs available on a website so that they can be easily downloaded. Matrix Algorithms, Volume II: Eigensystems is an excellent treatise on eigensystems. It should be purchased by all libraries and any individuals who make extensive use of computations in eigensystems.

A nonoverlapping domain decomposition method for nonconforming finite element problems

A nonoverlapping domain decomposition method for nonconforming finite element problems of second order partial differential equations is developed and analyzed. In particular, its convergence is demonstrated and convergence rate is estimated. The method is based on a Robin boundary condition as its transmission condition together with a derivative-free transmission data updating technique on the interfaces. The method is directly presented to finite element problems without introducing any Lagrange multipliers. The method can be naturally applied to general multi-subdomain decompositions and implemented on parallel machines with local communications. The method also allows choosing subdomains very flexibly, which can be even as small as an individual element. Therefore, the method can be regarded as a bridge connecting between direct methods and iterative methods for linear systems. Finally, some numerical experiments are also presented to demonstrate the effectiveness of the method.

Efficient and reliable iterative methods for linear systems

The approximate solutions in standard iteration methods for linear systems Ax= b, with A an n by n nonsingular matrix, form a subspace. In this subspace, one may try to construct better approximations for the solution x. This is the idea behind Krylov subspace methods. It has led to very powerful and efficient methods such as conjugate gradients, GMRES, and Bi-CGSTAB. We will give an overview of these methods and we will discuss some relevant properties from the user's perspective view. The convergence of Krylov subspace methods depends strongly on the eigenvalue distribution of A, and on the angles between eigenvectors of A. Preconditioning is a popular technique to obtain a better behaved linear system. We will briefly discuss some modern developments in preconditioning, in particular parallel preconditioners will be highlighted: reordering techniques for incomplete decompositions, domain decomposition approaches, and sparsified Schur complements.

Preconditioned Aor Iterative Methods for Linear Systems

In this paper we present the comparison analysis between preconditioned AOR method and AOR method under wider conditions than those in recent papers. Some numerical examples are given to illustrate our assertion.

Convergence of Pseudocontractions and Applications to Two-stage and Asynchronous Multisplitting for Singular M-Matrices

Pseudocontractions, which are generalizations of paracontractions in the linear case, are introduced in this paper in order to study the convergence of nonstationary iterative methods for linear systems in which the coefficient matrices are singular M-matrices. A general convergence theorem for pseudocontractions is developed. This theorem is used to analyze the convergence of two nonstationary parallel iterations: two-stage multisplitting iterations and asynchronous multisplitting iterations for singular M-matrices, without other contractivity conditions on the iteration matrices.

The convergence conditions of the psd iterative method for linear systems

In this paper, the PSD iterative method was proposed by Evans and Missirlis [4], for solving a large nonsingular system of linear equations Ax=b A general necessary condition for con-vergence of the PSD iterative method is obtained. The convergence theorems of the PSD iterative method are established under the condition that the coefficient matrix A is an H-matrix, our theorems improve and extend some known results.

Numerical analysis for applied science

Approximation of Functions. Direct Methods for Linear Systems. Solution of Nonlinear Equations. Iterative Methods for Linear Systems. Eigenvalue Problems. Numerical Integration. Ordinary Differential Equations. Difference Methods for PDEs. Introduction to Finite Elements. Appendices. Index.