Showing papers on "Matrix (mathematics) published in 1977"

An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix

Abstract: A particular class of regular splittings of not necessarily symmetric M-matrices is proposed. If the matrix is symmetric, this splitting is combined with the conjugate-gradient method to provide a fast iterative solution algorithm. Comparisons have been made with other well-known methods. In all test problems the new combination was faster than the other methods.

Realizability of Reynolds-Stress Turbulence Models

Abstract: It is shown that certain existing phenomenological models of turbulence in terms of differential equations for the Reynolds stresses Rαβ=〈u′α u′β〉 do not guarantee realizable solutions. The known realizability conditions are Rαβ⩾0 for α=β and Rαβ2⩾Rαα Rββ for α≠β. A stronger requirement is that the matrix Rαβ be positive semi‐definite. This implies three conditions like non‐negative eigenvalues or non‐negative principal invariants. Conditions are given which must be satisfied by the model itself in order to guarantee realizable solutions for any realizable initial and boundary conditions. Some means are proposed which can be used to change the existing models into realizable ones.

Matrix Computation for Engineers and Scientists

Abstract: A book for engineers who wish to use matrices in digital computation. The main theme is matrix numerical analysis, particularly the solution of linear simultaneous equations and eigen-value problems. Selected applications have been introduced and certain features of computer implementation have been discussed.

Solution of symmetric linear complementarity problems by iterative methods

Abstract: A unified treatment is given for iterative algorithms for the solution of the symmetric linear complementarity problem: $$Mx + q \geqslant 0, x \geqslant 0, x^T (Mx + q) = 0$$ , whereM is a givenn×n symmetric real matrix andq is a givenn×1 vector. A general algorithm is proposed in which relaxation may be performed both before and after projection on the nonnegative orthant. The algorithm includes, as special cases, extensions of the Jacobi, Gauss-Seidel, and nonsymmetric and symmetric successive over-relaxation methods for solving the symmetric linear complementarity problem. It is shown first that any accumulation point of the iterates generated by the general algorithm solves the linear complementarity problem. It is then shown that a class of matrices, for which the existence of an accumulation point that solves the linear complementarity problem is guaranteed, includes symmetric copositive plus matrices which satisfy a qualification of the type: $$Mx + q > 0 for some x in R^n $$ . Also included are symmetric positive-semidefinite matrices satisfying this qualification, symmetric, strictly copositive matrices, and symmetric positive matrices. Furthermore, whenM is symmetric, copositive plus, and has nonzero principal subdeterminants, it is shown that the entire sequence of iterates converges to a solution of the linear complementarity problem.

Numerical Computation of the Matrix Exponential with Accuracy Estimate

Abstract: This paper presents and analyzes an algorithm for computing the exponential of an arbitrary $n \times n$ matrix. Diagonal Pade table approximations are used in conjunction with several techniques for reducing the norm of the matrix. An important feature of the algorithm is that an estimate for the minimum number of digits accurate in the norm of the computed exponential matrix is returned to the user. In obtaining this estimate, several interesting results concerning rounding errors and Pade approximations are presented.

A design technique for linear multivariable feedback systems

Abstract: A systematic approach is developed for the design of linear multivariable feedback control systems based on a manipulation of the set of frequency-conscious eigenvalues and eigenvectors of an open-loop transfer-function matrix. The key idea behind the approach used is that of an approximately-commutative controller. An algorithm for approximation of a frame of complex vectors by a frame of real vectors is developed and plays a basic role in the systematic design approach. An example, based on industrial plant data, is given showing how the design method is used.

New results in 2-D systems theory, part I: 2-D polynomial matrices, factorization, and coprimeness

Abstract: During recent years, linear system theory has intensively been applied in estimation and control. At the same time, image processing has attracted increasing interest and attempts have been made to extend the techniques of systems theory to multidimensional problems, among others, by Bose, Attasi, Givone and Roesser, and Mitra. Part I of our results is centered around polynomial descriptions of systems. The notion of minimality in connection with state space requires the concept of coprimeness of 2-D polynomial matrices. For this purpose, we have extended the existing 1-D results on greatest common right divisor (GCRD) extraction, Sylvester resultants, matrix fraction descriptions (MFD) to the 2-D case. In addition we have results that appear to be unique for multidimensional problems such as existence and uniqueness of so-called "primitive factorizations" and existence of general factorizations. Part II will appear in a companion paper presenting results on a comparison between the different state space models that have been proposed, using what we consider to be proper definitions of state, controllability and observability and their relation to minimality of 2-D systems. We also represent new implementations of 2-D transfer functions using a minimal number of dynamic elements.

Inverses of Band Matrices and Local Convergence of Spline Projections

Abstract: It is shown that the size of the entries in the inverse of a band matrix can be bounded in terms of the norm of the matrix, the norm of its inverse and the bandwidth In many cases this implies that the entries of the inverse decay to zero exponentially as they move away from the diagonal These results are used to obtain local convergence theorems for some spline projections

Robustness of linear quadratic state feedback designs in the presence of system uncertainty

Abstract: The well-known stabilizing property of linear quadratic state feedback (LQSF) design is used to obtain a quantitative measure of the robustness of LQSF designs in the presence of perturbations. Bounds are obtained for allowable nonlinear, time-varying perturbations such that the resulting closed-loop system remains stable. The special case of linear, time-invariant perturbations is also treated. The bounds are expressed in terms of the weighting matrices in a quadratic performance index and the corresponding positive definite solution of the algebraic matrix Riccati equation, and are easy to compute for any given LQSF design. A relationship is established between the perturbation bounds and the dominant eigenvalues of the closed-loop optimal system model. Some interesting asymptotic properties of the bounds are also discussed. An autopilot for the flare control of the Augmentor Wing Jet STOL Research Aircraft (AWJSRA) is designed, based on LQSF theory, and the results presented in this paper. The variation of the perturbation bounds to changes in the weighting matrices in the LQSF design is studied by computer simulations, and appropriate weighting matrices are chosen to obtain a reasonable bound for perturbations in the system matrix and at the same time meet the practical constraints for the flare maneuver of the AWJSRA. Results from the computer simulation of a satisfactory autopilot design for the flare control of the AWJSRA are presented.

Exterior finite elements for 2-dimensional field problems with open boundaries

Abstract: Electric- and magnetic-field problems with boundaries at infinity are treated in finite-element terms by constructing an element to model an extremely large annulus surrounding the region of interest. A simple recursion technique is employed to generate the matrix representing the annular region. All nodes are eliminated from the external element except those on its inner surface, so that the final matrix is no larger than that required to describe the region of interest only. The method is simpler to program and requires less computing effort than boundary-integral techniques. It has been tested by solving several 2-dimensional magnetostatic and electrostatic problems and comparing the results with analytic solutions. The method can be applied to any 2-dimensional field problem bounded by a large empty region in which the field satisfies Laplace's equation.

Eigenvalue-generalized eigenvector assignment with state feedback

Abstract: In a recent paper [1], a characterization has been given for the class of all closed-loop eigenvector sets which can be obtained with a given set of distinct closed-loop eigenvalues. This note extends these results to characterize the class of generalized eigenvector chains which can be obtained with a given set of nondistinct eigenvalues. Included is an algorithm for computing a feedback matrix which gives the selected closed-loop eigenvalues and generalized eigenvector chains. Although there are limitations on the Jordan structure of the closed-loop system, this algorithm allows one to realize any "allowable" closed-loop Jordan configuration.

A Method for Calculating the Frequency-Dependent Properties of Microstrip Discontinuities

Abstract: A method is described for calculating the dynamical (frequency-dependent) properties of various microstrip discontinuities such as unsymmetrical crossings, T junctions, right-angle bends, impedance steps, and filter elements. The method is applied to an unsymmetrical T junction with three different linewidths. Using a waveguide model with frequency-dependent parameters, a field matching method proposed by Kuhn is employed to compute the scattering matrix of the structures. The elements of the scattering matrix calculated in this way differ from those derived from static methods by a higher frequency dependence, especially for frequencies near the cutoff frequencies of the higher order modes on the microstrip lines. The theoretical results are compared with measurements, and theory and experiment are found to correspond closely.

Screening of matrix suitable for immobilization of microbial cells

Abstract: To find a suitable matrix for immobilization of microbial cells, synthetic and natural polymers were screened. As a result,kappa-carrageenan,iota-carrageenan, furcellaran, sodium alginate, ethyl succinylated cellulose, succinylated zein, and 2-methyl-5-vinyl-pyridine-methylacrylate-methacrylic acid copolymer were studied. These polymers were induced to gel under mild conditions.Streptomyces phaeochromogenes cells having glucose isomerase activity were successfully immobilized in these polymer matrices. If a gelinducing reagent were added to a substrate solution, these gel matrices could be stabilized. The microbial cells did not leak out from the gel lattice. When these immobilized cells were treated with hardening reagents such as glutaraldehyde or tannins, the gel matrices were strengthened, and the glucose isomerase activity became stable for a long period even in the absence of gel-inducing reagents. Among these polymer matrices tested,kappa -carrageenan was most suitable for immobilization of microbial cells.

When is a Matrix Sign Stable.

Abstract: : A square real matrix A is called sign stable or qualitatively stable if stability is a property of every matrix having the same sign-pattern (negative, zero, positive) as A. A combinatorial characterization of this notion, announced earlier by the first author and used by the other authors in an efficient algorithm for testing sign stability, is established here. (Author)

Singular Values, Diagonal Elements, and Convexity

Abstract: Necessary and sufficient conditions are given for the existence of a matrix with prescribed singular values and prescribed elements in its main diagonal. Numerous related results are also given.

Ergodicity of age structure in populations with Markovian vital rates, III: Finite-state moments and growth rate; an illustration

Abstract: Leslie (1945) models the evolution in discrete time of a closed, single-sex population with discrete age groups by multiplying a vector describing the age structure by a matrix containing the birth and death rates. We suppose that successive matrices are chosen according to a Markov chain from a finite set of matrices. We find exactly the long-run rate of growth and expected age structure. We give two approximations to the variance in age structure and total population size. A numerical example illustrates the ergodic features of the model using Monte Carlo simulation, finds the invariant distribution of age structure from a linear integral equation, and calculates the moments derived here.

Spherical Matrix Distributions and a Multivariate Model

Abstract: SUMMARY We consider inference about the parameters of a multivariate linear model, in which the usual assumption of normality for the errors is replaced by a weaker assumption of spherical symmetry. Structural distributions and confidence regions are derived, and it is shown that inference about means is identical with that appropriate under normality, being based on a matrix generalization of "Studentization". Some relevant distribution theory is developed, the approach throughout being "densityfree".

Hamilton's Principle: Finite-Element Methods and Flexible Body Dynamics

Abstract: A variational formulation is given for the equations of motion for an unconstrained elastic body, and Hamilton's principle is used to derive the equations of motion and deformation of the body. Finite-element approximations are developed for these dynamical equations with respect to a body axis system satisfying the mean axis conditions. The free-body influence matrix for the body then is developed in terms of the finite-element model parameters.

Composantes cubiques normales des tenseurs spheriques

Abstract: The entire matrix which connects 'standard' to 'cubic normal' components of spherical tensors is defined. Numerical values of the matrix elements are given up to J = 15. The results may be used for proper or improper groups (SO(3), O(3) – O, Td, Oh) useful in atomic and molecular physics.

Application of graphical methods of spin algebras to limited CI approaches. I. Closed shell case

Abstract: The time independent diagrammatic technique based on the mathematical methods of quantum electrodynamics (second quantization, Wick's theorem, Feynman-like diagrams) is combined with graphical techniques of spin algebras to derive general expressions for the matrix elements of spin independent one- and two-particle operators between spin symmetry adapted ground, mono- and bi-excited configurations of a closed shell system. Two coupling schemes are considered for bi-excited states and their relative merits are discussed. Finally, the results are used to derive compact expressions for the coupling coefficients of the direct configuration interaction from molecular integrals (CIMI) method.

Vector-coupling approach to orbital and spin-dependent tableau matrix elements in the theory of complex spectra

Abstract: The power of the Young tableau scheme for labeling a complete spin-adapted basis set in the theory of complex spectra lies in one's ability to evaluate matrix elements of irreducible tensor operators directly in terms of the tableau labels and shapes. We show that the matrix-element rules stated by Harter for one-body operators can be easily derived from simple vector-coupling considerations. The graphical method of angular momentum analysis is used to derive closed-form expressions for the matrix elements of two-body operators. This study yields several interesting new relationships between spin-dependent operators and purely orbital operators.

Combined representation method for use in band-structure calculations: Application to highly compressed hydrogen

Abstract: A representation is described whose basis functions combine the important physical aspects of a finite set of plane waves with those of a set of Bloch tight-binding functions. The chosen combination has a particularly simple dependence on the wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ within the Brillouin zone, and its use in reducing the standard oneelectron band-structure problem to the usual secular equation has the advantage that the lattice sums involved in the calculation of the matrix elements are actually independent of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$. For systems with complicated crystal structures, for which the Korringa-Kohn-Rostoker, augmented-plane-wave, and orthogonalized-plane-wave methods are difficult to use, the present method leads to results with satisfactory accuracy and convergence. It is applied here to the case of compressed molecular hydrogen taken in a $\mathrm{Pa}3$ ($\ensuremath{\alpha}\ensuremath{-}\mathrm{nitrogen}$) structure for various densities but with mean interproton distance held fixed. The bands show a marked free-electron character above 5 to 6 times the normal density, and the overall energy gap is found to vanish at 9.15 times normal density. Within the approximations made, this represents an upper bound for the molecular density in the transition to the metallic state from an $\ensuremath{\alpha}\ensuremath{-}\mathrm{nitrogen}$ structure.