Non-recursive equivalent of the conjugate gradient method without the need to restart

A simple alternative to the conjugate gradient(CG) method is presented; this method is developed as a special case of the more general iterated Ritz method (IRM) for solving a system of linear equations. This novel algorithm is not based on conjugacy, i.e. it is not necessary to maintain overall orthogonalities between various vectors from distant steps. This method is more stable than CG, and restarting techniques are not required. As in CG, only one matrix-vector multiplication is required per step with appropriate transformations. The algorithm is easily explained by energy considerations without appealing to the A-orthogonality in n-dimensional space. Finally, relaxation factor and preconditioning-like techniques can be adopted easily.Comment: 9 page

Exact arithmetic as a tool for convergence assessment of the IRM-CG method

Using exact computer arithmetic, it is possible to determine the (exact) solution of a numerical model without rounding error. For such purposes, a corresponding system of equations should be exactly defined, either directly or by rationalisation of numerically given input data. In the latter case there is an initial round off error, but this does not propagate during the solution process. If this system is first exactly solved, then by the floating-point arithmetic, convergence of the numerical method is easily followed. As one example, IRM-CG, a special case of the more general Iterated Ritz method and interesting replacement for a standard or preconditioned CG, is verified. Further, because the computer demands and execution time grow enourmously with the number of unknowns using this strategy, the possibilities for larger systems are also provided.Comment: 12 page

The Deflated Relaxed Incomplete Cholesky CG method for use in a real-time ship simulator

Ship simulators are used for training purposes and therefore have to calculate realistic wave patterns around the moving ship in real time. We consider a wave model that is based on the variational Boussinesq formulation, which results in a set of partial differential equations. Discretization of these equations gives a large system of linear equations, that has to be solved each time-step. The requirement of real-time simulations necessitates a fast linear solver. In this paper we study the combination of the Relaxed Incomplete Cholesky preconditioner and subdomain deflation to accelerate the Conjugate Gradient method. We show that the success of this approach depends on the relaxation parameter. For low values of the relaxation parameter, e.g. the standard IC preconditioner, the deflation method is quite successfull. This is not the case for large values of the relaxation parameter, such as the Modified IC preconditioner. We give a theoretical explanation for this difference by considering the spectrum of the preconditioned and deflated matrices. Computational results for the wave model illustrate the expected convergence behavior of the Deflated Relaxed Incomplete Cholesky CG method. We also present promising results for the combination of the deflation method and the inherently parallel block-RIC preconditioner.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc

Cumulative index to NASA Tech Briefs, 1986-1990, volumes 10-14

Tech Briefs are short announcements of new technology derived from the R&D activities of the National Aeronautics and Space Administration. These briefs emphasize information considered likely to be transferrable across industrial, regional, or disciplinary lines and are issued to encourage commercial application. This cumulative index of Tech Briefs contains abstracts and four indexes (subject, personal author, originating center, and Tech Brief number) and covers the period 1986 to 1990. The abstract section is organized by the following subject categories: electronic components and circuits, electronic systems, physical sciences, materials, computer programs, life sciences, mechanics, machinery, fabrication technology, and mathematics and information sciences

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described