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At least 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

The Eigenvector-Eigenvalue Identity and other pragmatic topics in linear algebra for physicists

Diagonalization of an Hermitian matrix is a common task in physics. All of us have diagonalized 2x2 matrices but few have diagonalized a 3x3 matrix algebraically except in special simplifying cases. In this colloquium, I will discuss the mathematics and methods for diagonalizing small, but larger than 2x2 marices, and discuss the recently rediscovered Eigenvector-Eigenvalue identity. As an explicit, pragmatic example I will use the propagation of neutrino's propagating through matter which is inherently a 3x3 problem.

Parke, Stephen [Fermilab] (ORCID:0000000320286782)↗
NASA NTRS1988

Leapfrog variants of iterative methods for linear algebra equations

Two iterative methods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given.

Saylor, Paul E.↗
NASA NTRS1988

Computational strategies and improvements in the linear algebraic variational approach to rearrangement scattering

The computational steps in calculating quantum mechanical reactive scattering amplitudes by the L2 generalized Newton variational principle are discussed with emphasis on computational strategies and recent improvements that make the calculations more efficient. Special emphasis is placed on quadrature techniques, storage management strategies, use of symmetry, and boundary conditions. It is concluded that an efficient implementation of these procedures provides a powerful algorithm for the accurate solution of the Schroedinger equation for rearrangements.

Schwenke, David W.↗
NASA NTRS1994

A high-speed linear algebra library with automatic parallelism

Parallel or distributed processing is key to getting highest performance workstations. However, designing and implementing efficient parallel algorithms is difficult and error-prone. It is even more difficult to write code that is both portable to and efficient on many different computers. Finally, it is harder still to satisfy the above requirements and include the reliability and ease of use required of commercial software intended for use in a production environment. As a result, the application of parallel processing technology to commercial software has been extremely small even though there are numerous computationally demanding programs that would significantly benefit from application of parallel processing. This paper describes DSSLIB, which is a library of subroutines that perform many of the time-consuming computations in engineering and scientific software. DSSLIB combines the high efficiency and speed of parallel computation with a serial programming model that eliminates many undesirable side-effects of typical parallel code. The result is a simple way to incorporate the power of parallel processing into commercial software without compromising maintainability, reliability, or ease of use. This gives significant advantages over less powerful non-parallel entries in the market.

Boucher, Michael L.↗
NASA NTRS1990

Accuracy requirements of optical linear algebra processors in adaptive optics imaging systems

A ground-based adaptive optics imaging telescope system attempts to improve image quality by detecting and correcting for atmospherically induced wavefront aberrations. The required control computations during each cycle will take a finite amount of time. Longer time delays result in larger values of residual wavefront error variance since the atmosphere continues to change during that time. Thus an optical processor may be well-suited for this task. This paper presents a study of the accuracy requirements in a general optical processor that will make it competitive with, or superior to, a conventional digital computer for the adaptive optics application. An optimization of the adaptive optics correction algorithm with respect to an optical processor's degree of accuracy is also briefly discussed.

Downie, John D.↗
NASA NTRS2008

Improved Linear Algebra Methods for Redshift Computation from Limited Spectrum Data - II

Given photometric broadband measurements of a galaxy, Gaussian processes may be used with a training set to solve the regression problem of approximating the redshift of this galaxy. However, in practice solving the traditional Gaussian processes equation is too slow and requires too much memory. We employed several methods to avoid this difficulty using algebraic manipulation and low-rank approximation, and were able to quickly approximate the redshifts in our testing data within 17 percent of the known true values using limited computational resources. The accuracy of one method, the V Formulation, is comparable to the accuracy of the best methods currently used for this problem.

Foster, Leslie↗