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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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NASA NTRS2003

Eigensolver for a Sparse, Large Hermitian Matrix

A parallel-processing computer program finds a few eigenvalues in a sparse Hermitian matrix that contains as many as 100 million diagonal elements. This program finds the eigenvalues faster, using less memory, than do other, comparable eigensolver programs. This program implements a Lanczos algorithm in the American National Standards Institute/ International Organization for Standardization (ANSI/ISO) C computing language, using the Message Passing Interface (MPI) standard to complement an eigensolver in PARPACK. [PARPACK (Parallel Arnoldi Package) is an extension, to parallel-processing computer architectures, of ARPACK (Arnoldi Package), which is a collection of Fortran 77 subroutines that solve large-scale eigenvalue problems.] The eigensolver runs on Beowulf clusters of computers at the Jet Propulsion Laboratory (JPL).

Tisdale, E. Robert↗
NASA NTRS1993

Polar decomposition for attitude determination from vector observations

This work treats the problem of weighted least squares fitting of a 3D Euclidean-coordinate transformation matrix to a set of unit vectors measured in the reference and transformed coordinates. A closed-form analytic solution to the problem is re-derived. The fact that the solution is the closest orthogonal matrix to some matrix defined on the measured vectors and their weights is clearly demonstrated. Several known algorithms for computing the analytic closed form solution are considered. An algorithm is discussed which is based on the polar decomposition of matrices into the closest unitary matrix to the decomposed matrix and a Hermitian matrix. A somewhat longer improved algorithm is suggested too. A comparison of several algorithms is carried out using simulated data as well as real data from the Upper Atmosphere Research Satellite. The comparison is based on accuracy and time consumption. It is concluded that the algorithms based on polar decomposition yield a simple although somewhat less accurate solution. The precision of the latter algorithms increase with the number of the measured vectors and with the accuracy of their measurement.

Bar-Itzhack, Itzhack Y.↗
NASA NTRS1968

Molecular collisions. VIII.

Molecular collisions, formulating description as set of differential equations and developing eigenvectors and eigenvalues of Hermitian matrix associated with coupling

Curtiss, C. F.↗
NASA NTRS1988

Generalized Eigenvalues for pairs on heritian matrices

A study was made of certain special cases of a generalized eigenvalue problem. Let A and B be nxn matrics. One may construct a certain polynomial, P(A,B, lambda) which specializes to the characteristic polynomial of B when A equals I. In particular, when B is hermitian, that characteristic polynomial, P(I,B, lambda) has real roots, and one can ask: are the roots of P(A,B, lambda) real when B is hermitian. We consider the case where A is positive definite and show that when N equals 3, the roots are indeed real. The basic tools needed in the proof are Shur's theorem on majorization for eigenvalues of hermitian matrices and the interlacing theorem for the eigenvalues of a positive definite hermitian matrix and one of its principal (n-1)x(n-1) minors. The method of proof first reduces the general problem to one where the diagonal of B has a certain structure: either diag (B) = diag (1,1,1) or diag (1,1,-1), or else the 2 x 2 principal minors of B are all 1. According as B has one of these three structures, we use an appropriate method to replace A by a positive diagonal matrix. Since it can be easily verified that P(D,B, lambda) has real roots, the result follows. For other configurations of B, a scaling and a continuity argument are used to prove the result in general.

Rublein, George↗
NASA NTRS1989

A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications

In recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices appeared in the literature. We study restricted congruence transformation of Hermitian matrices M which, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain short proofs of most of the known inertia theorems and also derive some new results of this type. For some applications, a special class of almost unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form which, in particular, displays the eigenvalues of A. Finally, applications of this quasi-spectral decomposition to generalize inverses and Hermitian matrix pencils are discussed.

Freund, Roland W.↗
NASA NTRS2003

NASA Tech Briefs, November 2003

Topics covered include: Computer Program Recognizes Patterns in Time-Series Data; Program for User-Friendly Management of Input and Output Data Sets; Noncoherent Tracking of a Source of a Data-Modulated Signal; Software for Acquiring Image Data for PIV; Detecting Edges in Images by Use of Fuzzy Reasoning; A Timer for Synchronous Digital Systems; Prototype Parts of a Digital Beam-Forming Wide-Band Receiver; High-Voltage Droplet Dispenser; Network Extender for MIL-STD-1553 Bus; MMIC HEMT Power Amplifier for 140 to 170 GHz; Piezoelectric Diffraction-Based Optical Switches; Numerical Modeling of Nanoelectronic Devices; Organizing Diverse, Distributed Project Information; Eigensolver for a Sparse, Large Hermitian Matrix; Modified Polar-Format Software for Processing SAR Data; e-Stars Template Builder; Software for Acoustic Rendering; Functionally Graded Nanophase Beryllium/Carbon Composites; Thin Thermal-Insulation Blankets for Very High Temperatures; Prolonging Microgravity on Parabolic Airplane Flights; Device for Locking a Control Knob; Cable-Dispensing Cart; Foam Sensor Structures Would be Self-Deployable and Survive Hard Landings; Real-Gas Effects on Binary Mixing Layers; Earth-Space Link Attenuation Estimation via Ground Radar Kdp; Wedge Heat-Flux Indicators for Flash Thermography; Measuring Diffusion of Liquids by Common-Path Interferometry; Zero-Shear, Low-Disturbance Optical Delay Line; Whispering-Gallery Mode-Locked Lasers; Spatial Light Modulators as Optical Crossbar Switches; Update on EMD and Hilbert-Spectra Analysis of Time Series; Quad-Tree Visual-Calculus Analysis of Satellite Coverage; Dyakonov-Perel Effect on Spin Dephasing in n-Type GaAs; Update on Area Production in Mixing of Supercritical Fluids; and Quasi-Sun-Pointing of Spacecraft Using Radiation Pressure.

Source record↗
NASA NTRS1987

Model reduction for discrete bilinear systems

A model reduction method for discrete bilinear systems is developed which matches q sets of Volterra and covariance parameters. These parameters are shown to represent both deterministic and stochastic attributes of the discrete bilinear system. A reduced order model which matches these q sets of parameters is defined to be a q-Volterra covariance equivalent realization (q-Volterra COVER). An algorithm is presented which constructs a class of q-Volterra COVERs parameterized by solutions to a Hermitian, quadratic, matrix equation. The algorithm is applied to a bilinear model of a robot manipulator.

King, A. M.↗
NASA NTRS2010

Broken Ergodicity in MHD Turbulence

Ideal magnetohydrodynamic (MHD) turbulence may be represented by finite Fourier series, where the inherent periodic box serves as a surrogate for a bounded astrophysical plasma. Independent Fourier coefficients form a canonical ensemble described by a Gaussian probability density function containing a Hermitian covariance matrix with positive eigenvalues. The eigenvalues at lowest wave number can be very small, resulting in a large-scale coherent structure: a turbulent dynamo. This is seen in computations and a theoretical explanation in terms of 'broken ergodicity' contains Taylor s theory of force-free states. An important problem for future work is the case of real, i.e., dissipative flows. In real flows, broken ergodicity and coherent structure are still expected to occur in MHD turbulence at the largest scale, as suggested by low resolution simulations. One challenge is to incorporate coherent structure at the largest scale into the theory of turbulent fluctuations at smaller scales.

Shebalin, John V.↗
NASA NTRS1989

A fast, preconditioned conjugate gradient Toeplitz solver

A simple factorization is given of an arbitrary hermitian, positive definite matrix in which the factors are well-conditioned, hermitian, and positive definite. In fact, given knowledge of the extreme eigenvalues of the original matrix A, an optimal improvement can be achieved, making the condition numbers of each of the two factors equal to the square root of the condition number of A. This technique is to applied to the solution of hermitian, positive definite Toeplitz systems. Large linear systems with hermitian, positive definite Toeplitz matrices arise in some signal processing applications. A stable fast algorithm is given for solving these systems that is based on the preconditioned conjugate gradient method. The algorithm exploits Toeplitz structure to reduce the cost of an iteration to O(n log n) by applying the fast Fourier Transform to compute matrix-vector products. Matrix factorization is used as a preconditioner.

Pan, Victor↗
NASA NTRS1984

Wave normal directions of chorus near the equatorial source region

OGO 5 triaxial search coil magnetometer data are used to determine the wave propagation directions of postmidnight chorus in the near-equatorial region at L shells of 6 to 7. The methods used to estimate the wave normal directions involved minimum variance, the imaginary part of the cross-spectral matrix, the eigenvector of the Hermitian cross-spectral matrix, and a fitting of dispersion relations for one-wave and two-wave models to the cross-spectral matrix. Wave propagation at all frequencies within chorus tones is found to occur most frequently along the magnetic field with median and average cone angles of 9.1 deg and 12.2 deg, respectively. It is concluded that for waves propagating parallel to B, wave growth is maximum.

Goldstein, B. E.↗
NASA NTRS1973

Free vibration analysis of spinning structural systems.

This article presents an efficient digital computer procedure, along with the complete listing of the associated computer program, which may be conveniently utilized for the accurate solution of a wide range of practical eigenvalue problems. Important applications of the present work are envisaged in the natural frequency analysis of spinning structures discretized by the finite element technique, and in the determination of transfer functions associated with the dynamic blocks of control systems of spacecraft utilizing gas jets or reaction wheels for attitude control, as well as of spin-stabilized and dual-spin-stabilized satellites. The validity of the Sturm sequence property is first established for the related matrix formulation involving Hermitian and real symmetric, positive-definite matrices, both being usually of highly banded configuration. A numerically stable algorithm based on the Sturm sequence method is then developed which fully exploits the banded form of the associated matrices.

Gupta, K. K.↗
NASA NTRS1985

Parallel solution of closely coupled systems

An odd-even permutation and a nested dissection technique were used to circumvent the strong seriality of a system of closely coupled equations. The effect of transforming the n x n Hermitian definite positive matrix coefficient on the topology of Cholesky factors is discussed. A series of directed graphs is constructed in order to show the computational steps required for the odd-even permutation. Numerical expressions for the speed-up and efficiency of parallel N-processing techniques and sequential processing by a single computer are derived. Similar expressions are derived for the case of insufficient processing capacity. The application of the odd-even permutation to the ensemble class of computer architectures is demonstrated.

Utku, S.↗
NASA NTRS1989

A family of permutations for concurrent factorization of block tridiagonal matrices

The inherent strong seriality of closely coupled systems is circumvented by defining a family of permutations for reordering equation sets whose matrix of coefficients is Hermitian block tridiagonal. The authors show how these permutations can be used to achieve relatively high concurrency in the Cholesky factorization of banded systems at the expense of introducing limited extra computations due to fill-in terms in the factors. Directed graphs are developed for the concurrent factorization of the transformed matrix of coefficients by the Cholesky algorithm. Expressions for speedup and efficiency are derived in terms of parameters of the permutation, set of equations, and machine architecture.

Utku, Senol↗
NASA NTRS1995

Solution of the three-dimensional Helmholtz equation with nonlocal boundary conditions

The Helmholtz equation is solved within a three-dimensional rectangular duct with a nonlocal radiation boundary condition at the duct exit plane. This condition accurately models the acoustic admittance at an arbitrarily-located computational boundary plane. A linear system of equations is constructed with second-order central differences for the Helmholtz operator and second-order backward differences for both local admittance conditions and the gradient term in the nonlocal radiation boundary condition. The resulting matrix equation is large, sparse, and non-Hermitian. The size and structure of the matrix makes direct solution techniques impractical; as a result, a nonstationary iterative technique is used for its solution. The theory behind the nonstationary technique is reviewed, and numerical results are presented for radiation from both a point source and a planar acoustic source. The solutions with the nonlocal boundary conditions are invariant to the location of the computational boundary, and the same nonlocal conditions are valid for all solutions. The nonlocal conditions thus provide a means of minimizing the size of three-dimensional computational domains.

Hodge, Steve L.↗
NASA NTRS1975

Useful extremum principle for the variational calculation of matrix elements. II

Recent work (Gerjuoy et al., 1974) on variational principles for diagonal bound state matrix elements of arbitrary Hermitian operators is extended. In particular, it is shown that the previously derived minimum principle for the trial auxiliary function appearing in such variational principles can be constructed using a modified Hamiltonian possessing not heretofore recognized positive definite properties. Thus there is at least one alternative to the particular modified Hamiltonian on which the results of Gerjuoy et al. (1974) originally were based.

Gerjuoy, E.↗
NASA NTRS1988

Fast polar decomposition of an arbitrary matrix

The polar decomposition of an m x n matrix A of full rank, where m is greater than or equal to n, can be computed using a quadratically convergent algorithm. The algorithm is based on a Newton iteration involving a matrix inverse. With the use of a preliminary complete orthogonal decomposition the algorithm can be extended to arbitrary A. How to use the algorithm to compute the positive semi-definite square root of a Hermitian positive semi-definite matrix is described. A hybrid algorithm which adaptively switches from the matrix inversion based iteration to a matrix multiplication based iteration due to Kovarik, and to Bjorck and Bowie is formulated. The decision when to switch is made using a condition estimator. This matrix multiplication rich algorithm is shown to be more efficient on machines for which matrix multiplication can be executed 1.5 times faster than matrix inversion.

Higham, Nicholas J.↗
NASA NTRS1974

Useful extremum principle for the variational calculation of matrix elements

Variational principles are considered for the approximate evaluation of the diagonal matrix elements of an arbitrary known linear Hermitian operator. A method is derived that is immediately applicable to the variational determination of both the off-diagonal and diagonal matrix elements of normal and modified Green's functions.

Gerjuoy, E.↗
NASA NTRS1993

Application and development of the Schwinger multichannel scattering theory and the partial differential equation theory of electron-molecule scattering

One version of the multichannel theory for electron-target scattering based on the Schwinger variational principle, the SMC method, requires the introduction of a projection parameter. The role of the projection parameter a is investigated and it is shown that the principal-value operator in the SMC equation is Hermitian regardless of the value of a as long as it is real and nonzero. In a basis that is properly orthonormalizable, the matrix representation of this operator is also Hermitian. The use of such basis is consistent with the Schwinger variational principle because the Lippmann-Schwinger equation automatically builds in the correct boundary conditions. Otherwise, an auxiliary condition needs to be introduced, and Takatsuka and McKoy's original value of a is one of the three possible ways to achieve Hermiticity. In all cases but one, a can be uncoupled from the Hermiticity condition and becomes a free parameter. An equation for a based on the variational stability of the scattering amplitude is derived; its solution has an interesting property that the scattering amplitude from a converged SMC calculation is independent of the choice of a even though the SMC operator itself is a-dependent. This property provides a sensitive test of the convergence of the calculation. For a static-exchange calculation, the convergence requirement only depends on the completeness of the one-electron basis, but for a general multichannel case, the a-invariance in the scattering amplitude requires both the one-electron basis and the N plus 1-electron basis to be complete. The role of a in the SMC equation and the convergence property are illustrated using two examples: e-CO elastic scattering in the static-exchange approximation, and a two-state treatment of the e-H2 Chi(sup 1)Sigma(sub g)(+) yields b(sup 3)Sigma(sub u)(+) excitation.

Weatherford, Charles A.↗