Showing papers on "Matrix (mathematics) published in 1973"

Introduction to matrix computations

Abstract: Preliminaries Practicalities The Direct Solution of Linear Systems Norms, Limits, and Condition Numbers The Linear Least Squares Problem Eigenvalues and Eigenvectors The QR Algorithm The Greek Alphabet and Latin Notational Correspondents Determinants Rounding-Error Analysis of Solution of Triangular Systems and of Gaussian Elimination Of Things Not Treated Bibliography Index

Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth‐order wavefunctions

Abstract: A method is proposed to calculate the effect of configuration interaction by a Rayleigh‐Schrodinger perturbation expansion when starting from a multiconfigurational wavefunction. It is shown that a careless choice of H0 may lead to absurd transition energies between two states, at the first orders of the perturbation, even when the perturbation converges for both states. A barycentric defintion of H0 is proposed, which ensures the cancellation of common diagrams in the calculated transition energies. A practical iterative procedure is defined which allows a progressive improvement of the unperturbed wavefunction ψ0; the CI matrix restricted to a subspace S of strongly interacting determinants is diagonalized. The desired eigenvector ψ0 of this matrix is perturbed by the determinants which do not belong to S. The most important determinants in ψ1 are added to S, etc. The energy thus obtained after the second‐order correction is compared with the ordinary perturbation series where ψ0 is a single determinant...

The differentiation of pseudo-inverses and non-linear least squares problems whose variables separate.

Abstract: For given data (t, Yi), l, , m, we consider the least squares fit ofnonlinear models of the form It is shown that by defining the matrix {(0t)}i, qgj(0t; ti), and the modified functional r2(0t (lY O(0t)/(0t)yl)22, it is possible to optimize first with respect to the parameters 0t, and then to obtain, a posteriori, the optimal parameters . The matrix (0t) is the Moore-Penrose generalized inverse of O(t). We develop formulas for the Fr6chet derivative of orthogonal projectors associated with and also for /(0t), under the hypothesis that O(0t) is of constant (though not necessarily full) rank. Detailed algorithms are presented which make extensive use ofwell-known reliable linear least squares techniques, and numerical results and comparisons are given. These results are generalizations of those of H. D. Scolnik (20) and Guttman, Pereyra and Scolnik (9).

An Algorithm for Generalized Matrix Eigenvalue Problems.

Abstract: A new method, called the $QZ$ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the $QR$ algorithm, and reduces to it when $B = I$. Problems involving higher powers of $\lambda $ are also mentioned.

Binary Matrices in System Modeling

Abstract: This primarily tutorial paper on the use of binary matrices in system modeling also includes new material related to the initial development of such matrices. The decomposition of binary matrices into levels such that all feedback is contained within the levels is illustrated. A method for developing a binary matrix en route to a structural model of a system is outlined. The development procedure partitions the matrix on the basis of supplied data entries. Then the interconnections between subsystems are added. This procedure permits transitivity to be used in developing the matrix.

Algorithms for the nonlinear eigenvalue problem

Abstract: The following nonlinear eigenvalue problem is studied : Let $T(\lambda )$ be an $n \times n$ matrix, whose elements are analytical functions of the complex number $\lambda $. We seek $\lambda $ and vectors x and y, such that $T(\lambda )x = 0$, and $y^H T(\lambda ) = 0$.Several algorithms for the numerical solution of this problem are studied. These algorithms are extensions of algorithms for the linear eigenvalue problem such as inverse iteration and the $QR$ algorithm, and algorithms that reduce the nonlinear problem into a sequence of linear problems. It is found that this latter method can be extended into a global strategy, finding a complete basis of eigenvectors in the cases where it is proved that such a basis exists.Numerical tests, performed in order to compare the different algorithms, are reported, and a few numerical examples illustrating their behavior are given.

Propagation of acoustic surface waves in multilayers: A matrix description

Abstract: A matrix formalism is outlined for studying surface wave propagation in layered structures utilizing materials of arbitrary anisotropy, piezoelectricity, and conductivity. The analytical formulation reduces the equations describing the system to a first‐order matrix differential equation which is readily solved. The conceptual advantages and reduction of programming effort and computing time resulting from this approach are illustrated using as examples an infinite plate, a multilayer plate, a layered half‐space, and a free half‐space.

Tangent Stiffness in Plane Frames

Abstract: With reference to planar framed structures, a tangent stiffness matrix is derived that does not require any approximation beyond those used in the conventional beam-column theory. The matrix is given in such a form as to clearly separate the contributions of large rigid body displacements from elastic and locally nonlinear effects, and its numerical evaluation appears to be routine. Possible approximations, analogous to those used by previous investigators, are also examined.

Identification and Optimization of Aircraft Dynamics

Abstract: A technique is described for the design of an adaptive controller for multivariable systems and is based on recently developed methods for identification and optimization. An application of the method to a helicopter system with time-varying parameters is considered in detail. The response of the adaptive system is compared with the corresponding response of a system with a fixed controller and a system using optimal control. The comparison reveals the almost optimal character of the adaptive system. Nomenclature A = n X n, system matrix B = n X m, input matrix C = n X n, model matrix F = m X n, feedback matrix G — n X n, model matrix (estimate of A) H = n X m, model matrix (estimate of B) K = n x n, symmetric Riccati matrix P = n X n, symmetric positive definite matrix used in the model Q, S — n X n, symmetric positive semidefinite matrices of the performance index Qi = n X n, symmetric positive definite Lyapunov matrix R = m X m, symmetric positive definite matrix of the

A Monte-Carlo study

Abstract: The theory of the estimation methods discussed in Chapter II actually involves no difficulties at all. In fact, the matrix of observations of the explanatory variables X is of full rank and in such a case the least squares method is applied; though if matrix X is not of full rank it may be better to use the best linear P-unbiased estimators.

Computer-aided two-dimensional analysis of bipolar transistors

Abstract: A method for solving numerically the two-dimensional (2D) semiconductor steady-state transport equations is described. The principles of this method have been published earlier [1]. This paper discusses in detail the method and a number of considerable improvements. Poisson's equation and the two continuity equations are discretized on two networks of different rectangular meshes. The 2D continuity equations are approximated by a set of difference equations assuming that the hole and electron current density components along the meshlines are constant between two neighboring meshpoints in a way similar to that used by Gummel and Scharfetter [2] for the one-dimensional (1D) continuity equations. The resulting difference approximations have generally a much larger validity range than the conventional difference formulations where it is assumed that the change in electrostatic potential between two neighboring points is small compared with k T/q . Therefore, a much smaller number of meshpoints is necessary than for the conventional difference approximations. This reduces considerably the computation time and the required memory space. It will be shown that the matrix of the coefficients of this set of difference equations is always positive definite. This is an important property and guarantees convergence and stability of the numerical solution of the continuity equations. The way in which the difference approximations for the continuity equations are derived gives directly consistent expressions for the current densities that can be used for calculating the currents. In order to demonstrate the kind of solutions obtainable, steady-state results for a bipolar n-p-n silicon transistor are presented and discussed.

Optimization of k nearest neighbor density estimates

Abstract: Nonparametric density estimation using the k -nearest-neighbor approach is discussed. By developing a relation between the volume and the coverage of a region, a functional form for the optimum k in terms of the sample size, the dimensionality of the observation space, and the underlying probability distribution is obtained. Within the class of density functions that can be made circularly symmetric by a linear transformation, the optimum matrix for use in a quadratic form metric is obtained. For Gaussian densities this becomes the inverse covariance matrix that is often used without proof of optimality. The close relationship of this approach to that of Parzen estimators is then investigated.

Incremental Finite Element Matrices

Abstract: A common technique in geometrically nonlinear finite element analysis is to express the total potential in terms of Lagrangian displacement coordinates, differentiate the potential to obtain the equilibrium equations, and form the differentials of the equilibrium equations to obtain linear incremental equilibrium equations. The geometric nonlinearities in the strain-displacement equations give rise to incremental matrices in the preceding equations. The form of these matrices is not unique in the expression for the total potential. The paper presents expressions for incremental matrices that remain valid in the equilibrium equations and in the linear incremental equilibrium equations. The construction of such matrices is illustrated for truss elements, in-plane bending elements, membrane elements, and plate flexural elements. An examination of some of the recent literature indicates that some investigators have used inappropriate forms of these incremental matrices.

Matrix Calculus Operations and Taylor Expansions

Abstract: In problems of large dimensional complexities, matrix methods are frequently the favored mathematical tools. In this paper some extensions of matrix methods to calculus operations are introduced. Consistent array structural definitions are given for derivatives of matrix-valued functions with respect to matrices, for matrix differentials, and for matrix integrals, and some operational properties arising therefrom are detailed. Novel structures are developed for Taylor expansions of a matrix-valued function, which have some attractive features both for manipulative and for computational purposes.

Structure and design of linear model following systems

Abstract: This paper investigates the problem of designing a compensating control for a linear multivariable system so that the impulse response matrix of the resulting closed-loop system coincides with the impulse response matrix of a prespecified linear model; this is the model following problem. A new formulation of the problem is developed, and necessary and sufficient conditions for a solution to exist are given. An upper bound is determined for the number of integrators needed to construct the compensating control and, if the open-loop plant in question possesses a left-invertible transfer matrix, this bound is shown to be as small as possible. The relationship between the internal structure of a model following system and the model being followed is explained, and a description is given of the possible distributions of system eigenvalues which can be achieved while maintaining a model following configuration. This leads to a statement of necessary and sufficient conditions for the existence of a solution to the problem which results in a stable compensated system.

Generalized Inversion of Modified Matrices

Abstract: For an $m \times n$ complex matrix A and two columns, c and d, representations for the Moore–Penrose inverse of the matrix $A + cd^ * $ are given for all possible cases. Moreover, each representation involves only A, $A^\dag $, c, d, and their conjugate transposes.

Computing Maximum Likelihood Estimates for the Mixed A.O.V. Model Using the W Transformation

Abstract: The W transformation, a matrix transformation, is developed and applied for the mixed analysis of variance model to compute maximum likelihood estimates of the variance components and fixed parameters. This transformation not only eliminates the need for the explicit computation of the n × n inverse matrix H−l but permits handling the iterative calculations such that they do not depend upon n (the number of observations) in any way. Although not wedded to a particular numerical method, the W transformation is implemented in conjunction with a modified Newton-Raphson method in which variance components are restricted to being non-negative.

Direct construction of minimal bilinear realizations from nonlinear input-output maps

Abstract: The paper is concerned with the construction of bilinear state space descriptions from a prescribed nonlinear input-output map. The problem is shown to be equivalent to that of matching an infinite sequence of constant parameters which uniquely identifies the given map. Both the problems of requiring the matching over a finite number of terms of the sequence (partial realization problem) and over the whole sequence (complete realization problem) are treated. In both cases explicit existence criteria and an algorithm for finding minimal realizations are given. The approach is based on the introduction of a suitable infinite matrix formed with the input-output parameters, which can be considered as a generalization of the Hankel matrix usually considered in the realization theory of linear systems.

A review of the matrix Riccati equation

Abstract: As usual, R denotes the field of real numbers, R" stands for the n-dimensional vector space over R, a prime denotes the transpose of a matrix, an asterisk denotes the complex conjugate transpose of a matrix, and P _ Q means that P — Q is hermitian or real symmetric nonnegative matrix. Square brackets represent matrices composed of the symbols inside. In order to get a better motivation for the problems to be discussed we first pose the underlying physical problem. Given the linear, continuous-time, constant system

Inverse Problem of Linear Optimal Control

Abstract: Necessary and sufficient conditions are derived such that a multi-input, time-varying, linear state-feedback system minimizes a quadratic performance index (the inverse linear optimal control problem). A procedure for determining all such equivalent performance indices that yield the same feedback matrix is indicated.

Descriptions of the Polarization States of Vector Processes: Applications to ULF Magnetic Fields

Abstract: In recent years a wide variety of methods has been used to describe the polarization characteristics of ULF (.001 to 1 Hz) magnetic fields. This paper gives a detailed outline of some of the descriptions derived from the spectral matrices of n-variate stochastic processes. The matrices are expanded in three different, standard sets of matrices in order to add some simplification to the interpretation of the polarizations. One set is composed of n-squared trace-orthogonal, hermitean matrices and leads directly to a generalization of the Stokes parameters and the degree of polarization for n-variate processes. The second set is developed from the dyad expansion, which in particular cases is analogous to the spectral decomposition of the matrix. The third set is composed of n commuting idempotent matrices and proves to be the most useful set when the stochastic process is not strictly polarized.

A matrix technique for constructing slip‐line field solutions to a class of plane strain plasticity problems

Abstract: A procedure is described for solving plane strain rigid perfect-plasticity problems which lead to linear integral equations. The problem of finding the initial characteristic (slip-line), from which the complete field can be constructed, is reduced to a simple matrix inversion. Although the form of this matrix will depend on the particular problem concerned, it will be expressible in terms of a few fundamental matrices which occur in all problems of this type. The properties of these basic matrices and FORTRAN subroutines for assimilating them and for performing the corresponding linear transformations are given in detail. In illustration the procedure is applied to a drawing and to a strip rolling problem.

Optimal Load Flow Solution Using the Hessian Matrix

Abstract: The rapid convergence that Newton's method possesses, by use of the Jacobian matrix, has led to an investigation of using a higher order matrix, the Hessian, for an even faster convergence. It turns out that this approach unifies the fields of nonlinear programming methods and Newton based methods. The load flow problem can be defined as the solution of a system of simultaneous equations fi(x)= O, i= l, ..., n. It can be shown the Newton's method proceeds in a direction that minimizes F=?fi(x)2. The Hessian load flow also minimizes F by assuming that it is a quadratic function, such that the linearizations become HD?=-g, where the Hessian H is the matrix of the second partials of F and the vector g is the gradient of F. The optimal load flow problem can be formulated by including some additional terms in F so that a single algorithm, based on the Hessian, essentially solves both the normal and the optimal load flow problems. An interesting aspect of the method is that an existing Newton's program can be updated to a Hessian program quite simply. The H matrix is somewhat less sparse than the corresponding Jacobian but enough so that sparse techniques should be used. Furthermore, the Hessian can be completely obtained from the Jacobian, thus avoiding extra explicit function evaluations in the program. The paper presents enough details of the method for an implementation of a computer program. Numerical examples are given and compared with Newton's method results.