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At least 271 records · Page 15
All sourcesNASA NTRSDOE OSTI
NASA NTRS1973

Sparse matrix methods based on orthogonality and conjugacy

A matrix having a high percentage of zero elements is called spares. In the solution of systems of linear equations or linear least squares problems involving large sparse matrices, significant saving of computer cost can be achieved by taking advantage of the sparsity. The conjugate gradient algorithm and a set of related algorithms are described.

Lawson, C. L.↗
NASA NTRS1972

Elimination on sparse symmetric systems of a special structure.

Consideration of the problem of finding a permutation of rows and columns and an algorithm for solving ordered systems of linear algebraic equations with sparse matrices having a certain regular structure. Two approaches to the solution of this problem, in which the sparsity is used to some extent, are outlined. One of them is a very general approach where optimal (or nearly optimal) ordering is sought and the algorithm for solving the ordered system treats the matrix element by element to perform only necessary operations. The other approach involves the use of band matrices. After comparing these two approaches, a third approach is then suggested which involves the use of pipe matrices, and a means of ordering the rows and columns to obtain this type of matrix is presented. Examples of matrices reordered by the proposed procedure are cited.

Segethova, J.↗
NASA NTRS1976

An algorithm for reducing the bandwidth and profile of a sparse matrix

A new algorithm for reducing the bandwidth and profile of a sparse matrix is described. Extensive testing on finite element matrices indicates that the algorithm typically produces bandwidth and profile which are comparable to those of the commonly-used reverse Cuthill-McKee algorithm, yet requires significantly less computation time.

Gibbs, N. E.↗
NASA NTRS1977

Preserving sparseness in multivariate polynominal factorization

Attempts were made to factor these ten polynomials on MACSYMA. However it did not get very far with any of the larger polynomials. At that time, MACSYMA used an algorithm created by Wang and Rothschild. This factoring algorithm was also implemented for the symbolic manipulation system, SCRATCHPAD of IBM. A closer look at this old factoring algorithm revealed three problem areas, each of which contribute to losing sparseness and intermediate expression growth. This study led to effective ways of avoiding these problems and actually to a new factoring algorithm. The three problems are known as the extraneous factor problem, the leading coefficient problem, and the bad zero problem. These problems are examined separately. Their causes and effects are set forth in detail; the ways to avoid or lessen these problems are described.

Wang, P. S.↗
NASA NTRS1981

Electromagnetic backscattering from a sparse distribution of lossy dielectric scatterers

Electromagnetic backscattering from a sparse distribution of lossy dielectric particles having random orientation and position is studied. The paper begins by using the Foldy approximation to find an equation for the mean field. From this equation, an effective permittivity for the scattering medium is obtained. The correlation of the scattered field is found by employing the distorted Born approximation, i.e., particles embedded in the effective medium are assumed to be single scatterers. The above method is then used to find the backscattering coefficients from a leaf canopy. The leaf canopy is modeled by a half space of dielectric discs that are small in comparison to a wavelength. Numerical results show that the depolarized cross section is a sensitive function of leaf inclination angle statistics.

Lang, R. H.↗
NASA NTRS1982

Iterative algorithms for large sparse linear systems on parallel computers

Algorithms for assembling in parallel the sparse system of linear equations that result from finite difference or finite element discretizations of elliptic partial differential equations, such as those that arise in structural engineering are developed. Parallel linear stationary iterative algorithms and parallel preconditioned conjugate gradient algorithms are developed for solving these systems. In addition, a model for comparing parallel algorithms on array architectures is developed and results of this model for the algorithms are given.

Adams, L. M.↗
NASA NTRS1984

Adapting iterative algorithms for solving large sparse linear systems for efficient use on the CDC CYBER 205

Adapting and designing mathematical software to achieve optimum performance on the CYBER 205 is discussed. Comments and observations are made in light of recent work done on modifying the ITPACK software package and on writing new software for vector supercomputers. The goal was to develop very efficient vector algorithms and software for solving large sparse linear systems using iterative methods.

Kincaid, D. R.↗
NASA NTRS1984

Algorithms for solving large sparse systems of simultaneous linear equations on vector processors

Very efficient algorithms for solving large sparse systems of simultaneous linear equations have been developed for serial processing computers. These involve a reordering of matrix rows and columns in order to obtain a near triangular pattern of nonzero elements. Then an LU factorization is developed to represent the matrix inverse in terms of a sequence of elementary Gaussian eliminations, or pivots. In this paper it is shown how these algorithms are adapted for efficient implementation on vector processors. Results obtained on the CYBER 200 Model 205 are presented for a series of large test problems which show the comparative advantages of the triangularization and vector processing algorithms.

David, R. E.↗