Overview of Iterative Linear System Solver Packages

P SPARSLIBis a library of portable FORTRAN routines for sparse matrix compuations. The current thrust of the library is in iterative solution techniques. In this note we present the`accelerators' part of the library, which consists of the best known of Krylov subspace techniques. This iterative solution module is implemented in reverse communication mode so as to allow any preconditioned to be combined with the pacgake. In addition, this mechanism allows us to ensure portability, since the communication calls required in the iterative solution process are hidden in the dot product and the matrix-vector product and preconditioning operatins. P SPARSLIB 4 CGNR This algorithm is intended for solving linear systms as well as leastsquares problems. It consists of solving the linear system, A T Ax = A T b by a CG method. Since A T A is always positive semi-de nite, it is guaranteed, in theory, to always converge to a solution. CGNR may be a good approach for highly inde nite matrices. For example if the matrix is unitary, then it can solve the linear system in just one step, whereas most of the other Krylov subspace projection methods will typically converge slowly. For typical problems arising from the discretization of partial di erential equations, CGNR converges more slowly than CG or BCG and so this approach is not as popular in this particular context.

The main topic of this thesis is updating preconditioners for solving large sparse linear systems Ax = b by using Krylov iterative methods. Two interesting types of problems are considered. In the first one is studied the iterative solution of nonsingular, non-symmetric linear systems where the coefficient matrix A has a skewsymmetric part of low-rank or can be well approximated with a skew-symmetric low-rank matrix. Systems like this arise from the discretization of PDEs with certain Neumann boundary conditions, the discretization of integral equations as well as path following methods, for example, the Bratu problem and the Love's integral equation. The second type of linear systems considered are least squares (LS) problems that are solved by considering the solution of the equivalent normal equations system. More precisely, we consider the solution of modified and rank deficient LS problems. By modified LS problem, it is understood that the set of linear relations is updated with some new information, a new variable is added or, contrarily, some information or variable is removed from the set. Rank deficient LS problems are characterized by a coefficient matrix that has not full rank, which makes difficult the computation of an incomplete factorization of the normal equations. LS problems arise in many large-scale applications of the science and engineering as for instance neural networks, linear programming, exploration seismology or image processing. Usually, incomplete LU or incomplete Cholesky factorization are used as preconditioners for iterative methods. The main contribution of this thesis is the development of a technique for updating preconditioners by bordering. It consists in the computation of an approximate decomposition for an equivalent augmented linear system, that is used as preconditioner for the original problem. The theoretical study and the results of the numerical experiments presented in this thesis show the performance of the preconditioner technique proposed and its competitiveness compared with other methods available in the literature for computing preconditioners for the problems studied. To my family. A special feeling of gratitude to my wife Elisa Savoia and my daughter Lucía Guerrero. I also dedicate this thesis to my friends who have supported me throughout the process. To each professor I had during my education, in particular, my project coordinators José Marín, José Mas and Juana Cerdán who have been more than generous with their expertise and precious time spent with me for preparing this thesis. To the

Linear Algebra and its Applications, 1996

2012

The objective of this dissertation is the design and analysis of iterative methods for the numerical solution of large, sparse linear systems. This type of systems emerges from the discretization of Partial Differential Equations. Two special types of linear systems are studied. The first type deals with systems whose coefficient matrix is two cyclic whereas the second type studies the augmented linear systems. Initially, the Preconditioned Simultaneous Displacement (PSD) method, which is a generalized version of the Symmetric SOR (SSOR) method, is studied when the Jacobi iteration matrix is weakly cyclic and its eigenvalues are all real “real case” or all imaginary “imaginary case”. The first result is that the PSD method has better convergence rate than the SSOR method. In particular, in the “imaginary case” its convergence is increased by an order of magnitude compared to the SSOR method. In an attempt to further increase the convergence rate of the PSD method, more parameters we...

2000

Iterative methods are a popular way of solving linear systems of equation

2009

We describe PIM Parallel Iterative Methods a collection of Fortran routines to solve systems of linear equations on parallel computers using iterative methods A number of iterative methods for symmetric and nonsymmetric systems are avail able including Conjugate Gradients CG Bi Conjugate Gradients Bi CG Conjugate Gradients squared CGS the stabilised version of Bi Conjugate Gradients Bi CGSTAB the restarted stabilised version of Bi Conjugate Gradients RBi CGSTAB generalised min imal residual GMRES generalised conjugate residual GCR normal equation solvers CGNR and CGNE quasi minimal residual QMR with coupled two term recurrences transpose free quasi minimal residual TFQMR and Chebyshev acceleration The PIM routines can be used with user supplied preconditioners and left right or symmetric preconditioning are supported Several stopping criteria can be chosen by the user In this user s guide we present a brief overview of the iterative methods and algorithms available The use of PIM is...

Scientific research and essays

In this paper, new preconditioners for solving linear systems are developed and preconditioned accelerated overrelaxation method (AOR) is used for the systems. The improvement of convergence rate via using new preconditioners method also shown. A numerical example is also given to illustrate our results. 2000 Mathematics Subject Classifications: 65F10, 15A06 Key Words and Phrases: linear systems, preconditioner, AOR iterative method, spectral radius, Z-, M- matrix

Applied Mathematics and Mechanics, 2006

The preconditioned Gauss-Seidel type iterative method for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed. Also the optimal parameter is presented. Numerical results show that the proper choice of the preconditioner can lead to effective by the preconditioned Gauss-Seidel type iterative methods for solving linear systems.

2008 Eighth IEEE International Conference on Data Mining, 2008

Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g. , payment of royalties). Copies may be requested from IBM T.

Numerical method is the important aspects in solving real world problems that are related to mathematics, science, medicine, business are very few examples. Numerical method is the area related to mathematics and computer science which create, analysis and implements algorithm to numerically solve the system of linear equations. Numerical methods commonly involve an iterative method (as to find roots). They are now mostly used as preconditions for the popular iterative solvers. While it is difficult task solve as it takes a lot of time but it is an interesting part of Mathematics. In this paper the main emphasis on the beginners that how to iterate the solution of numerical to get appropriate results.