Preconditioners For Matrices Research Articles

The use of matrix visualization in algorithmic design

The use of matrix visualization in the design and development of numerical algorithms for supercomputers is discussed. Using color computer graphics, numerical analysts can gain new insights into algorithm behavior, which can then be used to design more efficient (parallel) numerical algorithms. The application of a matrix visualization tool, MatVu, in the design of algorithms from numerical linear algebra is the primary focus. Specific examples include the derivation of optimal preconditioning matrices for a conjugate gradient method, the design of parallel hybrid algorithms for solving the symmetric eigenvalue problem, the effects of operator splitting in the solution of incompressible Navier-Stokes equations, and the monitoring of Jacobian matrices associated with the application of Newton's method to a corresponding nonlinear system of equations.

Reduced storage matrix methods in stiff ODE systems

Stiff initial-value ODE methods (e.g. BDF) normally require the Jacobian matrix as part of a Newton-like iteration within each implicit time step, and require it to be formed and stored explicitly, whether the linear algebraic method used is direct or iterative. But, a Krylov subspace iterative linear system method, which involves the system matrix only in operator form, can be made part of an inexact Newton method within such a stiff ODE method in a matrix-free manner, requiring no explicit Jacobian matrix storage. Such combinations, using BDF methods, have been implemented with Arnoldi iteration, GMRES (the Generalized Minimum RESidual method), and CG (the Conjugate Gradient Method). Various practical matters (scaling, starting, stopping, etc.) are dealt with in the stiff ODE context. In the context of general nonlinear algebraic systems, we provide some theoretical foundation for the combined Newton-Krylov method by giving convergence results that include errors due to the difference quotient approximation to the linear operator. Earlier tests showed matrix-free methods to be quite effective, at least when the spectrum of the kproblem Jacobian is rather tightly clustered. To improve their robustness, we have added preconditioning, in an experimental solver called LSODPK, which we tested on ODE systems that arise from time-dependent PDE systems by the method of lines. Preconditioner matrices can be formed from the interaction or reaction terms, from the spatial transport terms, or both (as in operator splitting). The additional matrix storage can be reduced greatly by grouping the diagonal blocks in a natural way. The methods appear to be quite effective in improving both speed and storage economy over traditional stiff methods, and over matrix-free methods without preconditioning.

Applications of the Lanczos method

In this paper we are concerned with the application of the unsymmetric two sided Lanczos method to systems of first order differential equations and its associated problems; namely generalized eigenproblems and linear systems of equations. The unsymmetric Lanczos method is first used to generate two sets of vectors; the left and right Lanczos vectors. These Lanczos vectors are used directly in a method of weighted residual to reduce the differential equations to a small unsymmetric tridiagonal system. The advantage of this method is that the reduced system can be solved directly thus eliminating the usual computation of the complex eigenpairs. An algorithm is then derived by simplifying the two sided Lanczos method for systems of equations with a symmetric matrix pair. By appropriate choice of the starting vectors we obtain an implementation of the Lanczos method that is remarkably close to that in ref. [9], but generalized to the case with indefinite matrix coefficients. This modification results in a simple relation between the left and right Lanczos vectors, a symmetric tridiagonal, and a diagonal matrix. The system of differential equations can then be reduced to one with a symmetric tridiagonal and diagonal coefficient matrices. The modified algorithm is also suitable for eigenproblems with symmetric indefinite matrices and for solution of systems of equations using indefinite preconditioning matrices. This approach is used to evaluate the vibration response of a damped beam problem and a space mast structure with a symmetric damping matrix arising from velocity feedback control forces. In both problems, accurate solutions were obtained with as few as 20 Lanczos vectors.

Toward Efficient Implementation of Preconditioned Conjugate Gradient Methods On Vector Supercomputers

We consider large, sparse linear systems that result from the discretization of partial differential equations on regular and ir regular domains, and we focus on the ap plication of the preconditioned conjugate gradient (PCCG) method to the solution of such systems. More specifically, our goal is the efficient implementation of the PCCG method on vector supercomputers. The contribution to the above goal is made by the introduction of a data struc ture that can be effectively manipulated on vector machines, the utilization of precon ditioning matrices obtained by incomplete factorization with diagonal update sets, and the introduction of new numbering schemes for both regular and irregular grids.

An Algorithm for the Inversion of Finite Toeplitz Matrices

possesses properties (1.1), (1.2), and (1.3). In order to find the joint probability density function of (yo, yi , , yn), or of any n + 1 successive variates, it is necessary to invert the matrix Tn . In this paper, we derive an exact recursive procedure for the numerical inversion of an arbitrary positive definite Toeplitz matrix of finite order, which takes full advantage of the strong restrictions placed on its elements by (1.1), (1.2), and (1.3). The number of multiplications required for the inversion of an nth order Toeplitz matrix, using this procedure, is proportional to n2, rather than to n', as in the case of methods which are suitable for arbitrary Hermitian matrices. To the author's knowledge, this inversion algorithm is the first to be specifically designed to take advantage of the peculiar simplicity of the general Toeplitz matrix. In addition, the closing section of the paper is devoted to a statement of an algorithm for the inversion of lnon-Hernmitian matrices of the form (1.1).