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Showing papers on "Matrix (mathematics) published in 1974"


Journal ArticleDOI
TL;DR: Superstructure as mentioned in this paper is a general-purpose automatic atomic-structure program that uses multi-configuration type expansions to calculate term energies, intermediate-coupling energy levels, term coupling coefficients, and cascade coefficients.

781 citations


Journal ArticleDOI
TL;DR: In this paper, a new method for network analysis, the "force density method" is presented, which is based upon the force-length ratios or force densities which are defined for each branch of the net structure.

766 citations


Journal ArticleDOI
TL;DR: In this article, the distribution of the unit sphere in three-space is discussed and the maximum likelihood estimators for the diagonal shape and concentration matrix (Z$ and the orthogonal orientation matrix (M$) are derived.
Abstract: The distribution $\Psi(\mathbf{x}; Z, M) = \operatorname{const}. \exp(\mathrm{tr} (ZM^T \mathbf{xx}^T M))$ on the unit sphere in three-space is discussed. It is parametrized by the diagonal shape and concentration matrix $Z$ and the orthogonal orientation matrix $M. \Psi$ is applicable in the statistical analysis of measurements of random undirected axes. Exact and asymptotic sampling distributions are derived. Maximum likelihood estimators for $Z$ and $M$ are found and their asymptotic properties elucidated. Inference procedures, including tests for isotropy and circular symmetry, are proposed. The application of $\Psi$ is illustrated by a numerical example.

564 citations


Journal ArticleDOI
TL;DR: The methods are intimately based on the recurrence of matrix factorizations and are linked to earlier work on quasi-Newton methods and quadratic programming.
Abstract: This paper describes two numerically stable methods for unconstrained optimization and their generalization when linear inequality constraints are added. The difference between the two methods is simply that one requires the Hessian matrix explicitly and the other does not. The methods are intimately based on the recurrence of matrix factorizations and are linked to earlier work on quasi-Newton methods and quadratic programming.

338 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered the plane interaction problem for a circular elastic inclusion embedded in an elastic matrix which contains an arbitrarily oriented crack, using the existing solutions for the edge dislocations as Green's functions.
Abstract: The plane interaction problem for a circular elastic inclusion embedded in an elastic matrix which contains an arbitrarily oriented crack is considered. Using the existing solutions for the edge dislocations as Green's functions, first the general problem of a through crack in the form of an arbitrary smooth arc located in the matrix in the vicinity of the inclusion is formulated. The integral equations for the line crack are then obtained as a system of singular integral equations with simple Cauchy kernels. The singular behavior of the stresses around the crack tips is examined and the expressions for the stress-intensity factors representing the strength of the stress singularities are obtained in terms of the asymptotic values of the density functions of the integral equations. The problem is solved for various typical crack orientations and the corresponding stress-intensity factors are given.

232 citations


Journal ArticleDOI
Richard J Turyn1
TL;DR: A number of special Baumert-Hall sets of units, including an infinite class, are constructed here; these give the densest known classes of Hadamard matrices.

191 citations


Journal ArticleDOI
TL;DR: In this article, the expected value of a multiplicative performance criterion, represented by the exponential of a quadratic function of the state and control variables, is minimized subject to a discrete stochastic linear system with additive Gaussian measurement and process noise.
Abstract: The expected value of a multiplicative performance criterion, represented by the exponential of a quadratic function of the state and control variables, is minimized subject to a discrete stochastic linear system with additive Gaussian measurement and process noise. This cost function, which is a generalization of the mean quadratic cost criterion, allows a degree of shaping of the probability density function of the quadratic cost criterion. In general, the control law depends upon a gain matrix which operates linearly on the smoothed history of the state vector from the initial to the current time. This gain matrix explicitly includes the covariance of the estimation errors of the entire state history. The separation theorem holds although the certainty equivalence principle does not. Two special cases are of importance. The first occurs when only the terminal state is costed. A feedback control law, linear in the current estimate of the state, results where the feedback gains are functionally dependent upon the error covariance of the current state estimate. The second occurs if all the intermediate states are costed but there is no process noise except for an initial condition uncertainty. A feedback law results which depends not only upon the current dynamical state estimate but also on an additional vector which is path dependent.

158 citations


Journal ArticleDOI
TL;DR: In this article, the magnetic moments and GT-type β-decay matrix elements of nuclei were studied from the point of view of configuration mixing in terms of second order perturbation theory.

150 citations


Journal ArticleDOI
TL;DR: In this article, a large class of backward scattering matrix elements involving Δk ∼ ± 2kF vanish for fermions interacting with two-body attractive forces in one dimension.
Abstract: We show that a large class of backward‐scattering matrix elements involving Δk ∼ ± 2kF vanish for fermions interacting with two‐body attractive forces in one dimension. (These same matrix elements are finite for noninteracting particles and infinite for particles interacting with two‐body repulsive forces.) Our results demonstrate the possibility of persistent currents in one dimension at T = 0, and are a strong indication of a metal‐to‐insulator transition at T = 0 for repulsive forces. They are obtained by use of a convenient representation of the wave operator in terms of density‐fluctuation operators.

148 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that even if the eigenvalues of the system A -matrix A(t) of a linear time-varying system are independent of t and some of them have positive real parts, the system is asymptotically stable.
Abstract: An example is given to show that even if the eigenvalues of the system A -matrix A(t) of a linear time-varying system \dot{x}(t) = A(t)x(t) are independent of t and some of them have positive real parts, the system is asymptotically stable.

138 citations


Journal ArticleDOI
TL;DR: In this article, a theory of matrix differentiation is presented which uses the concept of a matrix of derivative operators, which allows matrix techniques to be used in both the derivation and the description of results.
Abstract: A theory of matrix differentiation is presented which uses the concept of a matrix of derivative operators. This theory allows matrix techniques to be used in both the derivation and the description of results. Several new operations and identities are presented which facilitate the process of matrix differentiation. The derivative theorems and new operations are then applied to the problem of determining optimal policies in a linear decision model with unknown coefficients, a problem which would be cumbersome if not impossible to solve without these theorems and operations.

Journal ArticleDOI
TL;DR: An improved algorithm requiring only 3.5 times the power of currently available fast algorithms to solve a set of linear equations with a non-Hermitian Toeplitz associated matrix.
Abstract: The solution of a set of m linear equations with a non-Hermitian Toeplitz associated matrix is considered. Presently available fast algorithms solve this set with 4m2 “operations” (an “operation” is defined here as a set of one addition and one multiplication). An improved algorithm requiring only 3m2 “operations” is presented.

Journal ArticleDOI
TL;DR: In this paper, sufficient conditions for an n by n matrix to be D-stable are surveyed and relations among the conditions are given, and the verifiability of the thirteen conditions cited is also discussed.


Journal ArticleDOI
C. M. Lee1
TL;DR: In this article, a method for calculating the same parameters by solving the many-electron Schr\"odinger equation for an atom within a limited spherical volume is presented, for Ar states with $J=1$ and odd parity, compared with data extracted earlier from an analysis of spectral data.
Abstract: Previous papers have expressed various experimental spectral data in terms of a single set of parameters (eigen-quantum-defects, transformation matrix, and excitation dipole moments). The parameters pertain to eigenstates of an electron-ion scattering matrix which represents only the effect of short-range interactions. This paper presents a method for calculating the same parameters by solving the many-electron Schr\"odinger equation for an atom within a limited spherical volume. Quantitative results, for Ar states with $J=1$ and odd parity, are compared with data extracted earlier from an analysis of spectral data, and are also used to reproduce the first 65 discrete line positions and their intensities, the positions and profiles of autoionization lines, and the branching ratio of the photoionization.


Journal ArticleDOI
TL;DR: In this paper, a connected account of matrix quadratic equations is given, and some new results and new proofs of known results are given, as well as new proofs for known results.
Abstract: Matrix quadratic equations have found the most diverse applications. The present article gives a connected account of their theory, and contains some new results and new proofs of known results.

Journal ArticleDOI
TL;DR: In this paper, the spectral inverse of a Toeplitz matrix A whose form is related to that of a circulant matrix is studied by describing the algebraic structure of the semigroup of all matrices commuting with a given matrix with distinct eigenvalues.

Journal ArticleDOI
TL;DR: The bidiagonalization algorithm is shown to be the basis of important methods for solving the linear least squares problem for large sparse matrices.
Abstract: An algorithm given by Golub and Kahan [2] for reducing a general matrix to bidiagonal form is shown to be very important for large sparse matrices. The singular values of the matrix are those of the bidiagonal form, and these can be easily computed. The bidiagonalization algorithm is shown to be the basis of important methods for solving the linear least squares problem for large sparse matrices. Eigenvalues of certain 2-cyclic matrices can also be efficiently computed using this bidiagonalization.

Journal ArticleDOI
TL;DR: In this paper, a circuit-theoretic treatment for thin-film optical waveguides using a class of real anisotropic and gyrotropic materials is presented based on the two-mode approximation in normal mode theory.
Abstract: A circuit‐theoretic treatment is presented for thin‐film optical waveguides using a class of real anisotropic and gyrotropic materials. The analysis is based on the two‐mode approximation in normal‐mode theory. The terminal behavior of those guides is described by the 2 × 2 transmission matrix and expressions for matrix elements of six systems that are introduced as canonical elements in circuit synthesis are derived. Properties of canonical elements are discussed with particular attention on their reciprocity. Using the transmission matrix, we can treat the design of thin‐film optical devices by a simple matrix operation familiar in conventional transmission‐line circuit synthesis. As a typical application the design of various nonreciprocal integrated‐optical devices is treated in detail. The desired response is synthesized by cascading selected canonical elements in an appropriate order. Examples include the gyrator, unidirectional mode converter, differential phase shifter, isolator, and circulator.

Journal ArticleDOI
TL;DR: In this article, a theory of comminution machines that includes the effects of classification and predicts machine contents was developed, and the relation between models of this form and energy consumption laws was investigated.

Journal ArticleDOI
TL;DR: A unified continued fraction theory which can be considered as a generalized feedback theory is established and a multiple cycle model consisting of many feedback constant matrices and many feed forward integral matrices is constructed.
Abstract: A unified continued fraction theory which can be considered as a generalized feedback theory is established. A multiple cycle model consisting of many feedback constant matrices and many feed forward integral matrices is constructed. The outer feedback constant matrix corresponds to the first partial quotient matrix of the overall matrix continued fraction, whereas the outer forward integral matrix is corresponding to the second quotient matrix. Therefore, the partial quotient matrices of the continued fraction and the feedback and feed forward matrices of the system diagram are one-to-one correspondence in order. The influence of each matrix on the performance of the entire system depends on its position. The outer ones are much more important than the inner ones. A reduction model can be obtained by discarding certain inner matrices.

Journal ArticleDOI
TL;DR: In this paper, the determinant and inverse of the covariance matrix of a set of n consecutive observations on a mixed autoregressive moving average process are given for the general autoregression process of order p (n? p), and for the first order mixed auto-regression process.
Abstract: Expressions are obtained for the determinant and inverse of the covariance matrix of a set of n consecutive observations on a mixed autoregressive moving average process. Explicit formulae for the inverse of this matrix are given for the general autoregressive process of order p (n ? p), and for the first order mixed autoregressive moving average process.

Journal ArticleDOI
Peter M. Kogge1
TL;DR: It is shown that if the recurrence function f has associated with it two other functions that satisfy certain composition properties, then it can be constructed elegant and efficient parallel algorithms that can compute all N elements of the series in time proportional to ⌈log2N⌉.
Abstract: An mth-order recurrence problem is defined as the computation of the sequence x1,, xN, where x1 = f(ai, xi-1,,xi-m), and ai, is some vector of parameters This paper investigates general algorithms for solving such problems on highly parallel computers We show that if the recurrence function f has associated with it two other functions that satisfy certain composition properties, then we can construct elegant and efficient parallel algorithms that can compute all N elements of the series in time proportional to ⌈log2N⌉ The class of problems having this property includes linear recurrences of all orders- both homogeneous and inhomogeneous, recurrences involving matrix or binary quantities, and various nonlinear problemsin volving operations such as computation with matrix inverses, exponentiation, and modulo division

Journal ArticleDOI
TL;DR: The problem of finding the optimal, constant output feedback matrix for linear time-invariant multivariable systems with quadratic cost is reconsidered and simple formulae for the gradient matrix are developed and used in a Fletcher-Powell-Davidon algorithm.
Abstract: The problem of finding the optimal, constant output feedback matrix for linear time-invariant multivariable systems with quadratic cost is reconsidered. Simple formulae for the gradient matrix are developed and used in a Fletcher-Powell-Davidon algorithm. Computational results are presented.

Journal ArticleDOI
TL;DR: In this article, a comprehensive theory of the matrix linear equation $AX + XB = C$ is presented, where the equation is viewed as a vector equation in the vector space of all $m \times n$ matrices.
Abstract: A comprehensive theory of the matrix linear equation $AX + XB = C$ is presented. The equation is viewed as a vector equation $LX = C$ in the vector space of all $m \times n$ matrices. As the main result, two necessary and sufficient conditions for a solution to exist and a general form of all solutions are established.


Journal ArticleDOI
TL;DR: In this article, a parameterization of the nuclear S -matrix in terms of the K-matrix, employing a random matrix model for the latter, and assuming no direct reactions, numerically calculate elastic and inelastic compound nucleus cross sections.

Journal ArticleDOI
TL;DR: The surprising result is obtained that it may be beneficial to compute an unsymmetric factorization of a symmetric matrix under certain sparsity conditions.
Abstract: We consider the solution of linear equations involving a sparse coefficient matrix having a triangular factorization. In addition to the usual triangular factorization, we consider block factorizations, where the diagonal blocks of the factors are not necessarily triangular. We show that under certain sparsity conditions, these alternate factorizations may require fewer arithmetic operations and less storage. In particular, we obtain the surprising result that it may be beneficial to compute an unsymmetric factorization of a symmetric matrix.

Journal ArticleDOI
TL;DR: The transition matrix formulation of acoustic scattering given previously by Waterman [J. Acoust. Soc. Am. 45, 1417 (1969)] is extended to the case of an arbitrary number of scatterers in this paper.
Abstract: The transition matrix formulation of acoustic scattering given previously by Waterman [J. Acoust. Soc. Am. 45, 1417 (1969)] is extended to the case of an arbitrary number of scatterers. The resulting total transition matrix is expressed in terms of the individual transition matrices and in terms of functions which describe a translation of the origin for the spherical (and cylindrical) wave solutions of Helmholtz equation. Explicit formulas are given for the case of two and three scatterers and the (finite) iteration scheme for the general case is described. Some numerical calculations concerning some aspects of the scattering from two spheres are also reported.