Showing papers on "Matrix (mathematics) published in 1972"

Anharmonic force constant calculations

Abstract: The relationship of the anharmonic force constants in curvilinear internal coordinates to the observed vibration-rotation spectrum of a molecule is reviewed. A simplified method of setting up the required non-linear coordinate transformations is described: this makes use of an L tensor, which is a straightforward generalization of the L matrix used in the customary description of harmonic force constant calculations. General formulae for the L tensor elements, in terms of the familiar L matrix elements, are presented. The use of non-linear symmetry coordinates and redundancies are described. Sample calculations on the water and ammonia molecules are reported.

Correlation Matrix Memories

Abstract: A new model for associative memory, based on a correlation matrix, is suggested. In this model information is accumulated on memory elements as products of component data. Denoting a key vector by q(p), and the data associated with it by another vector x(p), the pairs (q(p), x(p)) are memorized in the form of a matrix {see the Equation in PDF File} where c is a constant. A randomly selected subset of the elements of M xq can also be used for memorizing. The recalling of a particular datum x(r) is made by a transformation x(r)=M xq q(r). This model is failure tolerant and facilitates associative search of information; these are properties that are usually assigned to holographic memories. Two classes of memories are discussed: a complete correlation matrix memory (CCMM), and randomly organized incomplete correlation matrix memories (ICMM). The data recalled from the latter are stochastic variables but the fidelity of recall is shown to have a deterministic limit if the number of memory elements grows without limits. A special case of correlation matrix memories is the auto-associative memory in which any part of the memorized information can be used as a key. The memories are selective with respect to accumulated data. The ICMM exhibits adaptive improvement under certain circumstances. It is also suggested that correlation matrix memories could be applied for the classification of data.

Interpretation of Inaccurate, Insufficient and Inconsistent Data

Abstract: Sumntary Many problems in physical science involve the estimation of a number of unknown parameters which bear a linear or quasi-linear relationship to a set of experimental data. The data may be contaminated by random errors, insufficient to determine the unknowns, redundant, or all of the above. This paper presents a method of optimizing the conclusions from such a data set. The problem is formulated as an ill-posed matrix equation, and general criteria are established for constructing an ‘ inverse ’ matrix. The ‘ solution ’ to the problem is defined in terms of a set of generalized eigenvectors of the matrix, and may be chosen to optimize the resolution provided by the data, the expected error in the solution, the fit to the data, the proximity of the solution to an arbitrary function, or any combination of the above. The classical ‘ least-squares ’ solution is discussed as a special case.

Methods for modifying matrix factorizations.

Abstract: In recent years several algorithms have appeared for modifying the factors of a matrix following a rank-one change. These methods have always been given in the context of specific applications and this has probably inhibited their use over a wider field. In this report several methods are described for modifying Cholesky factors. Some of these have been published previously while others appear for the first time. In addition, a new algorithm is presented for modifying the complete orthogonal factorization of a general matrix, from which the conventional QR factors are obtained as a special case. A uniform notation has been used and emphasis has been placed on illustrating the similarity between different methods.

A contribution to matrix quadratic equations

Abstract: The well-known matrix algebraic equation of the optimal control and filtering theory is considered. A necessary and sufficient condition for its solution to yield an optimal as well as asymptotically stable closed-loop system is given. The condition involves the concepts of stabilizability and detectability.

Characteristic modes for dielectric and magnetic bodies

Abstract: A theory of characteristic modes for material bodies, both with and without losses, is developed. For loss-free bodies, the modes consist of a set of real characteristic sources which diagonalize the generalized network matrix for the body, and a set of characteristic fields which diagonalize the scattering matrix. Most of the properties of these modes remain the same as those of the corresponding modes for perfectly conducting bodies. For lossy bodies, the corresponding modes have complex characteristic sources. However, in the lossy case there also exists a set of real characteristic sources which diagonalize the generalized network matrix, but their fields do not diagonalize the scattering matrix.

The Exact Finite Sample Properties of the Estimators of Coefficients in the Error Components Regression Models

Abstract: Wallace and Hussain (1969) considered the use of an error components regression model in the analysis of time series of cross-sections and developed an Aitken estimator of the coefficient vector based on an estimated variance-covariance matrix of error terms. In this paper, we have shown that under the set of assumptions adopted by Wallace and Hussain there are an infinite number of estimators which have the same asymptotic variancecovariance matrix as the Wallace-Hussain estimator and also that it is not possible to choose an estimator on the basis of asymptotic efficiency. We have developed an alternative estimator of the variance-covariance matrix of error terms and have used this estimator in developing a feasible "Aitken" type estimator for the coefficient vector. We have derived some small sample properties of this estimator and have compared them with those of other estimators of the coefficient vector.

Metal Matrix Composites

Abstract: Metal matrix composites consist of a metal or an alloy as the continuous matrix and a reinforcement that can be particle, short fiber or whisker, or continuous fiber. In this chapter, we first describe important techniques to process metal matrix composites, then we describe the interface region and its characteristics, properties of different metal matrix composites, and finally, we summarize different applications of metal matrix composites.

On the equinoctial orbit elements

Abstract: This paper investigates the equinoctial orbit elements for the two-body problem, showing that the associated matrices are free from singularities for zero eccentricities and zero and ninety degree inclinations. The matrix of the partial derivatives of the position and velocity vectors with respect to the orbit elements is given explicitly, together with the matrix of inverse partial derivatives, in order to facilitate construction of the matrizant (state transition matrix) corresponding to these elements. The Lagrange and Poisson bracket matrices are also given. The application of the equinoctial orbit elements to general and special perturbations is discussed.

Empirical Orthogonal Representation of Time Series in the Frequency Domain. Part I: Theoretical Considerations

Abstract: Difficulties in using conventional cross-spectrum analysis to explore atmospheric wave disturbances have indicated the need for some extension of the usual technique. It is suggested here that the eigenvectors of the cross-spectrum matrix be used for interpreting such data. The method is analogous to the use of empirical orthogonal functions applied to band-pass filtered time series. However, the eigenvectors of the cross-spectrum matrix contain additional information concerning phase which is not available from the eigenvectors of the covariance matrix. It is possible to generate a new set of time series which are mutually uncorrelated within a pre-selected frequency interval and which have the same combined variance in the frequency interval as the original set of time series. These new series are obtained by applying the eigenvectors of the cross-spectrum matrix to a set of complex time series involving the original time series and their time derivatives. The application and physical interpret...

Factor analysis by generalized least squares

Abstract: Aitken's generalized least squares (GLS) principle, with the inverse of the observed variance-covariance matrix as a weight matrix, is applied to estimate the factor analysis model in the exploratory (unrestricted) case. It is shown that the GLS estimates are seale free and asymptotically efficient. The estimates are computed by a rapidly converging Newton-Raphson procedure. A new technique is used to deal with Heywood cases effectively.

Invariant Expansion. II. The Ornstein‐Zernike Equation for Nonspherical Molecules and an Extended Solution to the Mean Spherical Model

Abstract: The Ornstein‐Zernike equation for fluids with nonspherical molecules obtained in a former publication [L. Blum and A. J. Torruella, J. Chem. Phys. 56, 303 (1972)] is written in coordinate space as a convolution matrix equation. A rather simple property of the angular coupling coefficients of our former work allows us to write the Ornstein‐Zernike equation in irreducible form, as a set of uncoupled matrix equations, of rather small size. A generalization of Baxter's form of the Ornstein‐Zernike equation to matrices allows us to write a formal solution to the mean spherical model of neutral hard spheres with almost arbitrary electrostatic multipoles. This is an extension of Wertheim's solution for dipoles [J. Chem. Phys. 55, 4291 (1971)]. The formal solution consists in showing that the direct correlation function inside the hard core is a polynomial in the interatomic distance r. The coefficients of the polynomials are obtained by solving a set of quadratic matrix equations. The class of potentials that ad...

A Class of Matrix Ensembles

Abstract: A class of random matrix ensembles is defined, with the purpose of providing a realistic statistical description of the Hamiltonian of a complicated quantum‐mechanical system (such as a heavy nucleus) for which an approximate model Hamiltonian is known. An ensemble of the class is specified by the model Hamiltonian H0, an observed eigenvalue distribution‐function r(E), and a parameter τ which may be considered to be a fictitious ``time.'' Each of H0, r(E), and τ may be chosen independently. The ensemble consists of matrices M which are obtained from H0 by an invariant random Brownian‐motion process, lasting for a time τ and tending to pull the eigenvalues of M toward the distribution r(E). For small τ the ensemble allows only small perturbations of H0. As τ → ∞, the ensemble tends to a stationary limit independent of H0 and depending on r(E) alone. The following quantitative results are obtained. (1) It is proved that the global eigenvalue distribution in the limit τ → ∞ becomes identical with the observe...

Electrical resistivity of conducting particles in an insulating matrix

Abstract: When conducting particles are added to a nonconducting matrix, conductivity increases abruptly at a critical concentration. The theory of infinite chains is used to predict the variation of conductivity with particle concentration. Factors which determine the critical concentration are discussed.

The measure of a matrix as a tool to analyze computer algorithms for circuit analysis

Abstract: The measure of a matrix is used to, first, bound solutions of ordinary differential equations, bound the computer solution by the backward Euler method, and bound the accumulated truncation error; second, to give conditions for the existence and uniqueness of a dc operating point; third, to determine a convergence region for the Newton-Raphson technique and establish its convergence properties. The unifying idea of the paper is the use of the measure of a matrix.

Design of optimal control systems with prescribed eigenvalues

Abstract: A quadratic performance index together with a set of prescribed closed-loop eigenvalues are considered as criteria for designing linear multi-variable control systems. A method is developed for finding the weighting matrix elements that correspond to the given eigenvalues. The method is completely general as to system structure and is easy to implement computationally. It is also shown how the method may be used to design optimal estimators with prescribed eigenvalues. Several examples are presented to illustrate the use of the method.

Sparse matrices and their applications

Abstract: Symposium on Sparse Matrices and Their Applications.- Computational Circuit Design.- Eigenvalue Methods for Sparse Matrices.- Sparse Matrix Approach to the Frequency Domain Analysis of Linear Passive Electrical Networks.- Some Basic Technqiues for Solving Sparse Systems of Linear Equations.- Vector and Matrix Variability Type in Sparse Matrix Algorithms.- Linear Programming.- The Partitioned Preassigned Pivot Procedure (P4).- Modifying Triangular Factors of the Basis in the Simplex Method.- Partial Differential Equations.- A New Iterative Procedure for the Solution of Sparse Systems of Linear Difference Equations.- Block Eliminations on Finite Element Systems of Equations.- Application of the Finite Element Method to Regional Water Transport Phenomena.- On the Use of Fast Methods for Separable Finite Difference Equations for the Solution of General Elliptic Problems.- Special Topics.- Application of Sparse Matrices to Analytical Photogrammetry.- Generalized View of a Data Base.- Combinatorics and Graph Theory.- Several Strategies for Reducing the Bandwidth of Matrices.- GRAAL - A Graph Algorithmic Language.- The Role of Partitioning in the Numerical Solution of Sparse Systems.

A Fast Computer Method for Matrix Transposing

Abstract: A method is given for transposition of 2n×2n data matrices, larger than available high-speed storage. The data should be stored on an external storage device, allowing direct access. The performance of the algorithm depends on the size of the main storage, which at least should hold 2n+1 points. In that case the matrix has to be read in and written out n times.

Condition of finite element matrices generated from nonuniform meshes.

Abstract: N a previous Note1 it has been shown (see also Refs. 2 and 3) that the spectral condition number Cn(K) of the global (stiffness) matrix K arising from a uniform mesh of finite elements (or of finite differences) discretization can be expressed by Cn(K) = cNes2m where 2m is the order of the differential equation and c a coefficient independent of Nes, the number of elements per side, but dependent on the order of the interpolation polynomials inside the element. This condition is "natural", since it is inherently associated with the approximation of the continuous problem by the discrete (algebraic) one. Nonuniform meshes of finite elements introduce many additional factors which may adversely affect the condition of the system. It is the purpose of this Note to describe a technique for establishing bounds on the condition number for irregular meshes of finite elements. With these bounds it becomes possible to study the effect of element geometry, the order of interpolation functions and other intrinsic and discretization parameters on Cn(K) and to isolate the factors that may lead to ill-conditioning. The matrix K is termed ill-conditioned when \Q~sCn(K) = 1, where s denotes the number of decimals in the computer. The bounds on Cn(K) are expressed in terms of the extremal eigenvalues of the element matrices. Since the element matrices are of restricted size, derivation of the bounds on Cn(K) as a function of the discretization parameters become straightforward for any problem and any element. Particular attention is focused on the possibility of improving the condition of the matrix by scaling. Bounds on the Extremal Eigenvalues

Compensation Methods for Network Solutions by Optimally Ordered Triangular Factorization

Abstract: The compensation theorem is applied in conjunction with ordered triangular factorization of the nodal admittance matrix to simulate the effect of changes in the passive elements of the network on the solution of a problem without changing the factorization. The scheme includes network elements with mutual impedances. Compared with impedance matrix methods which are ordinarily used for power system applications, this method, which permits exploitation of matrix sparsity, always requires less computer storage and, with few exceptions, is much faster.

Suggested Modification of the P-α Model for Porous Materials

Abstract: We suggest that the relationship between the pressure P in a porous material and the average matrix pressure Pm in the material should be P=Pm/α, where α is the ratio of the specific volumes of the porous material and the matrix material.

Oblique rotation to a partially specified target

Abstract: This paper presents an iterative procedure for rotating a factor matrix obliquely to a least-squares fit to a target matrix which need not be fully specified.

Comments on mictomagnetism

Abstract: The observed magnetic behavior of mictomagnetic alloys is reviewed. A model for the magnetic structure of such alloys, which assumes a random spin-glass matrix with magnetic clusters, is discussed in the light of magnetic, ESR, and Mossbauer spectroscopic results.

Polyhedral Sets Having a Least Element.

Abstract: For a fixedm × n matrixA, we consider the family of polyhedral setsX b ={x|Ax ≥ b}, b ∈ R m , and prove a theorem characterizing, in terms ofA, the circumstances under which every nonemptyX b has a least element. In the special case whereA contains all the rows of ann × n identity matrix, the conditions are equivalent toA T being Leontief. Among the corollaries of our theorem, we show the linear complementarity problem always has a unique solution which is at the same time a least element of the corresponding polyhedron if and only if its matrix is square, Leontief, and has positive diagonals.