Product form of inverses of sparse matrices and related topics Progress report, 1 Jun. 1965 - 30 Nov. 1967
Summaries of publications and reports generated in connection with studies on sparse matrix inverse product forms
Tewarson, R. P.↗
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Time-dependent treatment of scattering. II - Novel integral equation approach to quantum wave packets
The novel wave-packet propagation scheme presented is based on the time-dependent form of the Lippman-Schwinger integral equation and does not require extensive matrix inversions, thereby facilitating application to systems in which some degrees of freedom express the potential in a basis expansion. The matrix to be inverted is a function of the kinetic energy operator, and is accordingly diagonal in a Bessel function basis set. Transition amplitudes for various orbital angular momentum quantum numbers are obtainable via either Fourier transform of the amplitude density from the time to the energy domain, or the direct analysis of the scattered wave packet.
Sharafeddin, Omar A.↗
On some representations of generalized inverses.
Unitary matrix representing generalized inverses, proving weak method of steepest descent for least squares solution of equations
Tewarson, R. P.↗
Computer programs for the solution of systems of linear algebraic equations
FORTRAN subprograms for the solution of systems of linear algebraic equations are described, listed, and evaluated in this report. Procedures considered are direct solution, iteration, and matrix inversion. Both incore methods and those which utilize auxiliary data storage devices are considered. Some of the subroutines evaluated require the entire coefficient matrix to be in core, whereas others account for banding or sparceness of the system. General recommendations relative to equation solving are made, and on the basis of tests, specific subprograms are recommended.
Sequi, W. T.↗
Matrix differentiation formulas
A compact differentiation technique (without using indexes) is developed for scalar functions that depend on complex matrix arguments which are combined by operations of complex conjugation, transposition, addition, multiplication, matrix inversion and taking the direct product. The differentiation apparatus is developed in order to simplify the solution of extremum problems of scalar functions of matrix arguments.
Usikov, D. A.↗
SMI adaptive antenna arrays for weak interfering signals
The performance of adaptive antenna arrays in the presence of weak interfering signals (below thermal noise) is studied. It is shown that a conventional adaptive antenna array sample matrix inversion (SMI) algorithm is unable to suppress such interfering signals. To overcome this problem, the SMI algorithm is modified. In the modified algorithm, the covariance matrix is redefined such that the effect of thermal noise on the weights of adaptive arrays is reduced. Thus, the weights are dictated by relatively weak signals. It is shown that the modified algorithm provides the desired interference protection.
Gupta, Inder J.↗
A direct algorithm for solution of incompressible three-dimensional unsteady Navier-Stokes equations
A direct, implicit, numerical solution algorithm, with second-order accuracy in space and time, is constructed for the three-dimensional unsteady incompressible Navier-Stokes equations formulated in terms of velocity and vorticity, using generalized orthogonal coordinates to achieve the accurate solution of complex viscous flow configurations. A numerically stable, efficient, direct inversion procedure is developed for the computationally intensive divergence-curl elliptic velocity problem. This overdetermined partial differential operator is first formulated as a uniquely determined, nonsingular matrix-vector problem; this aspect of the procedure is a unique feature of the present analysis. The three-dimensional vorticity-transport equation is solved by a modified factorization technique which completely eliminates the need for any block-matrix inversions and only scalar tridiagonal matrices need to be inverted. The method is applied to the test problem of the three-dimensional flow within a shear-driven cubical box. Coherent streamwise vortex structures are observed within the steady-state flow at Re = 100.
Osswald, G. A.↗
Analysis of Modified SMI Method for Adaptive Array Weight Control
An adaptive array is used to receive a desired signal in the presence of weak interference signals which need to be suppressed. A modified sample matrix inversion (SMI) algorithm controls the array weights. The modification leads to increased interference suppression by subtracting a fraction of the noise power from the diagonal elements of the covariance matrix. The modified algorithm maximizes an intuitive power ratio criterion. The expected values and variances of the array weights, output powers, and power ratios as functions of the fraction and the number of snapshots are found and compared to computer simulation and real experimental array performance. Reduced-rank covariance approximations and errors in the estimated covariance are also described.
Dilsavor, Ronald Louis↗
Schur Complement Factorization and Parallel O (Log N) Algorithms for Computation of Operational Space Mass Matrix and Its Inverse
In this paper new factorization techniques for computation of the Operational Space Mass Matrix and its inverse are developed.
Schur simple physical interpretations highly effic↗
An experimental SMI adaptive antenna array simulator for weak interfering signals
An experimental sample matrix inversion (SMI) adaptive antenna array for suppressing weak interfering signals is described. The experimental adaptive array uses a modified SMI algorithm to increase the interference suppression. In the modified SMI algorithm, the sample covariance matrix is redefined to reduce the effect of thermal noise on the weights of an adaptive array. This is accomplished by subtracting a fraction of the smallest eigenvalue of the original covariance matrix from its diagonal entries. The test results obtained using the experimental system are compared with theoretical results. The two show a good agreement.
Dilsavor, Ronald S.↗
Spherical earth gravity and magnetic anomaly analysis by equivalent point source inversion
To facilitate geologic interpretation of satellite elevation potential field data, analysis techniques are developed and verified in the spherical domain that are commensurate with conventional flat earth methods of potential field interpretation. A powerful approach to the spherical earth problem relates potential field anomalies to a distribution of equivalent point sources by least squares matrix inversion. Linear transformations of the equivalent source field lead to corresponding geoidal anomalies, pseudo-anomalies, vector anomaly components, spatial derivatives, continuations, and differential magnetic pole reductions. A number of examples using 1 deg-averaged surface free-air gravity anomalies of POGO satellite magnetometer data for the United States, Mexico, and Central America illustrate the capabilities of the method.
Von Frese, R. R. B.↗
Aircraft electric field measurements: Calibration and ambient field retrieval
An aircraft locally distorts the ambient thundercloud electric field. In order to determine the field in the absence of the aircraft, an aircraft calibration is required. In this work a matrix inversion method is introduced for calibrating an aircraft equipped with four or more electric field sensors and a high-voltage corona point that is capable of charging the aircraft. An analytic, closed form solution for the estimate of a (3 x 3) aircraft calibration matrix is derived, and an absolute calibration experiment is used to improve the relative magnitudes of the elements of this matrix. To demonstrate the calibration procedure, we analyze actual calibration date derived from a Lear jet 28/29 that was equipped with five shutter-type field mill sensors (each with sensitivities of better than 1 V/m) located on the top, bottom, port, starboard, and aft positions. As a test of the calibration method, we analyze computer-simulated calibration data (derived from known aircraft and ambient fields) and explicitly determine the errors involved in deriving the variety of calibration matrices. We extend our formalism to arrive at an analytic solution for the ambient field, and again carry all errors explicitly.
Koshak, William J.↗
Reduction of Raman Spectroscopy Data for H2-CO2-Air Tubular Flame Measurements
An approach of reducing spontaneous Raman spectroscopy data, based on the matrix inversion method which logically removes undesired sources of the Raman scattering signals including cross-talk due to the signals from other species, is discussed. The approach contains extensive calibration of temperature-dependent system response. The Raman spectra of several cold and hot calibration flames and their resulting calibration matrix are elaborated. The structure of H2-CO2-air tubular flames, in terms of major species (CO2, O2, N2, H2O, and H2) concentrations and temperature, was successfully measured with the optical technique. The Raman spectra data used in this study are taken from a uniquely designed experimental apparatus – a tubular burner and Raman spectroscopy system at Vanderbilt University. The characteristic uncertainties for chemical species are approximately ±2% by mole fraction for hot products and ±0.5% for room temperature reactants. The approach described here for hydrogen/air tubular flames may be adapted to hydrocarbon/air tubular flames when other necessary major species, for example, CH4 and CO for methane/air tubular flame, are considered in the whole process.
Raman Spectroscopy↗
SMI adaptive antenna arrays for weak interfering signals
The performance of adaptive antenna arrays is studied when a sample matrix inversion (SMI) algorithm is used to control array weights. It is shown that conventional SMI adaptive antennas, like other adaptive antennas, are unable to suppress weak interfering signals (below thermal noise) encountered in broadcasting satellite communication systems. To overcome this problem, the SMI algorithm is modified. In the modified algorithm, the covariance matrix is modified such that the effect of thermal noise on the weights of the adaptive array is reduced. Thus, the weights are dictated by relatively weak coherent signals. It is shown that the modified algorithm provides the desired interference protection. The use of defocused feeds as auxiliary elements of an SMI adaptive array is also discussed.
Gupta, I. J.↗
Three-dimension heat transfer program
An IBM 7094, three dimensional, heat transfer program for satellite application is discussed. The program may be applied to all phases of spacecraft life, from launch to orbital quasi-steady state. Launch phase heating includes inputs by radiation from the hot fairing and free molecule flow heating after nose cone ejection. Orbital heat fluxes are accounted for by direct sunlight, earth-reflected sunlight, and earth-emitted input to the satellite. Internal heat exchange by conduction and radiation is determined, provided the appropriate conductances, radiation view factors, and effective emittances are known. A simplified analysis of an active thermal controlled spacecraft is incorporated into the program. One or more surfaces may contain shutter systems which have optical properties specified as a function of shutter opening. The program evaluates the feasibility of any configuration based upon maintaining an internal component within its tolerable temperature limits. The program employs a matrix inversion subroutine (Gauss Elimination Method) to solve the heat balance equations.
COMPUTER PROGRAM↗
On computing the fixed-point probability vector of ergodic transition matrices.
Fixed point probability vector of ergodic /cyclic/ transition matrices by matrix inversion technique
Decell, H. P.↗
Least squares and pseudoinversion
Matrix inversion and least squares fit procedures for postflight orbit and trajectory analysis
Lefferts, E. J.↗
The selection of spacecraft optical coatings for temperature control.
Computer optimization of spacecraft optical coatings for temperature control, using finite element analysis and matrix inversion
Costello, F. A.↗