The product form of inverses of sparse matrices and graph theory.
Geometric interpretation of product form of inverse applied to sparse matrices in linear programming, using graph theory
LINEAR PROGRAMMING↗
SEARCH · Engineering Papers
Results for “matrix algebra”
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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On the factorization of secular equations by group theory.
Explicit invariant forms of factor equations for factorization of secular equations by group theory
QUANTUM MECHANICS↗
Analytical Methods in Vibrations.
Book on analytical methods in vibrations covering general mathematical formulations of common features of various vibrating systems
STATISTICAL ANALYSIS↗
On the decoupling of multivariable systems.
Necessary and sufficient conditions for decoupling time-invariant linear multivariable system by state variable feedback, discussing transfer matrix consequences
MATRIX ALGEBRA↗
Methods for injection-error analysis and their comparison.
Statistical techniques for missile injection error analysis, discussing direct and adjoint methods
STATISTICAL ANALYSIS↗
A test for the sign of the second variation.
Determining sign of second variation in optimal control theory
MATRIX ALGEBRA↗
Matrix methods and automation in structural engineering
The partial differential equations of motion of shell structures subject to arbitrary loads may be extremely difficult or even impossible to solve. Since the digital computer is now an available tool for the structural engineer, current research has been directed toward methods which involve matrix formulation of large systems of algebraic equations and matrix solutions for discrete elements rather than the solution of the partial differential equations. These methods require rapid and accurate computer solutions. Recognizing the accuracy problems inherent in working with large matrices, a comprehensive survey of available computer programs was performed for large matrix inversion and for eigenvalue and eigenvector solutions under a Research Grant from the National Aeronautics and Space Administration and the techniques are developed in this paper for using these programs most efficiently for structural applications. The contents of the paper include: automation of matrix compilation, methods of very large matrix inversion and solution of simultaneous equation, and techniques for finding eigenvalues and vectors. In addition, a finite element stiffness matrix approach developed at the Denver Research Institute for both plates and shells subject to arbitrary dynamic loads is described as it was instrumented with complete automation on the digital computer.
Anita S West↗
Optimal control of spin stabilized spacecraft with telescoping appendages
The control of a spin-stabilized spacecraft consisting of a rigid central hub and one or two movable offset telescoping booms (with end masses) is considered. The equations of rotational motion are linearized about either of two desired final states. A control law for the boom and mass position is sought such that a quadratic cost functional involving the weighted components of angular velocity plus the control is minimized when the final time is unspecified and involves the solution of the matrix Riccati algebraic equation. For three-axis control more than one offset boom (orthogonal to each other) is required. For two-axis control with a single boom offset from a symmetrical hub, an analytic solution is obtained; when this system is used for nutation decay the time constant is one order of magnitude smaller than previously achieved using non-optimal control logic. For the general case results are obtained numerically.
Bainum, P. M.↗
Optimal control of spin-stabilized spacecraft with telescoping appendages
The control of a spin-stabilized spacecraft consisting of a rigid central hub and one or two movable offset telescoping booms (with end masses) is considered. The equations of rotational motion are linearized about either of two desired final states. A control law for the boom end mass position is sought such that a quadratic cost functional involving the weighted components of angular velocity plus the control is minimized when the final time is unspecified and involves the solution of the matrix Riccati algebraic equation. For three axis control more than one offset boom (orthogonal to each other) is required. For two-axis control with a single boom offset from a symmetrical hub, an analytic solution is obtained; when this system is used for nutation decay the time constant is one order of magnitude smaller than previously achieved using nonoptimal control logic. For the general case results are obtained numerically.
Bainum, P. M.↗
Optical systolic solutions of linear algebraic equations
The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.
Neuman, C. P.↗
Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review
A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances.
Kennedy, Christopher A.↗
Algebraic methods for the solution of some linear matrix equations
The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.
Djaferis, T. E.↗
Software for Fault-Tolerant Matrix Multiplication
Formal Linear Algebra Recovery Environment is a computer program for high-performance, fault-tolerant matrix multiplication. The program is based on an extension of the prior theory and practice of fault-tolerant matrix matrix multiplication of the form C = AB. This extension provides low-overhead methods for detecting errors, not only in C, but also in A and/or B. These methods enable the detection of all errors as long as, in a given case, only one entry in A, B, or C is corrupted. The program also provides for following a low-overhead rollback approach to correct errors once detected. Results of computational experiments have demonstrated that the methods implemented in this program work well in practice while imposing an acceptably low level of overhead, relative to high-performance matrix-multiplication methods that do not afford fault tolerance.
Katz, Daniel↗
A Matrix-Free Approach to Smoothed Aggregation Algebraic Multigrid.
Abstract not provided.
Harper, Graham↗
Research on the application of a decoupling algorithm for structure analysis
The mathematical theory for decoupling mth-order matrix differential equations is presented. It is shown that the decoupling precedure can be developed from the algebraic theory of matrix polynomials. The role of eigenprojectors and latent projectors in the decoupling process is discussed and the mathematical relationships between eigenvalues, eigenvectors, latent roots, and latent vectors are developed. It is shown that the eigenvectors of the companion form of a matrix contains the latent vectors as a subset. The spectral decomposition of a matrix and the application to differential equations is given.
Denman, E. D.↗
Electro-optical processor for optimal control
An iterative optical processor has been developed for applications in the optimal control of advanced sensor systems. The processor is designed for the realization of the Richardson algorithm on bipolar data, using as input a linear array of LEDs. The usefulness of the processor has been demonstrated by the solution of the linear quadratic regulator problem for the optimal control signals of the F100 turbofan engine. In this case study, the algebraic Riccati equation matrix was solved by the use of a modified Kleinman algorithm along with the Richardson algorithm applied to a system of linear algebraic equations. Preliminary experimental results demonstrate the gradual convergence of the processor.
Casasent, D.↗
Improved solution for ill-conditioned algebraic equations by epsilon decomposition
Matrix eigenvalue theory is presently used to examine the source of ill-conditioning in linear algebraic equations; the approach highlights the critical role played by the zero and near-zero eigenvalues and corresponding eigenvectors of poorly-conditioned systems. Insights derived from this approach are used to improve the recently developed epsilon-decomposition (E-D) solution procedure. The efficiency of E-D is significant for large matrices possessing small rank deficiency.
Ojalvo, Irving U.↗
Atomic data for opacity calculations. X - Oscillator strengths and photoionisation cross sections for O III
Energy levels and electric dipole radiative transitions were determined for O III. The wavefunctions for the bound and continuum states were derived by solving coupled integrodifferential equations in the close-coupling approximation with the aid of two methods, the R-matrix and linear algebraic methods. Configuration interaction wavefunctions are presented for eight states of the O IV target with configurations of 2s(x)sp(y) (x + y = 3). Oscillator strengths are calculated for the transitions between, and photoionization cross sections from, bound states of O III with configurations of 2s(x)2p(y)nl (for n of not greater than 10 and l of not greater than 3).
Luo, D.↗