Showing papers on "Matrix (mathematics) published in 1970"

A family of variable-metric methods derived by variational means

Abstract: A new rank-two variable-metric method is derived using Greenstadt's variational approach [Math. Comp., this issue]. Like the Davidon-Fletcher-Powell (DFP) variable-metric method, the new method preserves the positive-definiteness of the approximating matrix. Together with Greenstadt's method, the new method gives rise to a one-parameter family of variable-metric methods that includes the DFP and rank-one methods as special cases. It is equivalent to Broyden's one-parameter family [Math. Comp., v. 21, 1967, pp. 368-381]. Choices for the inverse of the weighting matrix in the variational approach are given that lead to the derivation of the DFP and rank-one methods directly. In the preceding paper [6], Greenstadt derives two variable-metric methods, using a classical variational approach. Specifically, two iterative formulas are developed for updating the matrix Hk, (i.e., the inverse of the variable metric), where Hk is an approximation to the inverse Hessian G-'(Xk) of the function being minimized.* Using the iteration formula Hk+1 = Hk + Ek to provide revised estimates to the inverse Hessian at each step, Greenstadt solves for the correction term Ek that minimizes the norm N(Ek) = Tr (WEkWEkJ) subject to the conditions

The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations

Abstract: This paper presents a more detailed analysis of a class of minimization algorithms, which includes as a special case the DFP (Davidon-Fletcher-Powell) method, than has previously appeared. Only quadratic functions are considered but particular attention is paid to the magnitude of successive errors and their dependence upon the initial matrix. On the basis of this a possible explanation of some of the observed characteristics of the class is tentatively suggested. PROBABLY the best-known algorithm for determining the unconstrained minimum of a function of many variables, where explicit expressions are available for the first partial derivatives, is that of Davidon (1959) as modified by Fletcher & Powell (1963). This algorithm has many virtues. It is simple and does not require at any stage the solution of linear equations. It minimizes a quadratic function exactly in a finite number of steps and this property makes convergence of this algorithm rapid, when applied to more general functions, in the neighbourhood of the solution. It is, at least in theory, stable since the iteration matrix H,, which transforms the jth gradient into the /th step direction, may be shown to be positive definite. In practice the algorithm has been generally successful, but it has exhibited some puzzling behaviour. Broyden (1967) noted that H, does not always remain positive definite, and attributed this to rounding errors. Pearson (1968) found that for some problems the solution was obtained more efficiently if H, was reset to a positive definite matrix, often the unit matrix, at intervals during the computation. Bard (1968) noted that H, could become singular, attributed this to rounding error and suggested the use of suitably chosen scaling factors as a remedy. In this paper we analyse the more general algorithm given by Broyden (1967), of which the DFP algorithm is a special case, and determine how for quadratic functions the choice of an arbitrary parameter affects convergence. We investigate how the successive errors depend, again for quadratic functions, upon the initial choice of iteration matrix paying particular attention to the cases where this is either the unit matrix or a good approximation to the inverse Hessian. We finally give a tentative explanation of some of the observed experimental behaviour in the case where the function to be minimized is not quadratic.

A general method for analysis of covariance structures

Abstract: SUMMARY It is assumed that observations on a set of variables have a multivariate normal distribution with a general parametric form of the mean vector and the variance-covariance matrix. Any parameter of the model may be fixed, free or constrained to be equal to other parameters. The free and constrained parameters are estimated by maximum likelihood. A wide range of models is obtained from the general model by imposing various specifications on the parametric structure of the general model. Examples are given of areas and problems, especially in the behavioural sciences, where the method may be useful. 1. GENERAL METHODOLOGY 11. The general model We consider a data matrix X = {xOq} of N observations on p response variables and the following model. Rows of X are independently distributed, each having a multivariate normal distribution with the same variance-covariance matrix E of the form

Dynamic mechanical properties of particulate-filled composites

Abstract: The relative shear moduli of composites containing glass spheres in a rubbery matrix obey the Mooney equation, analogous to the relative viscosity of similar suspensions in Newtonian liquids. However, when the matrix is a rigid epoxy, the relative shear moduli are less than what the Mooney equation predicts but greater than what the Kerner equation predicts. Relative moduli are less for rigid matrices than for rubbery matrices because (1) the modulus of the filler is not extremely greater compared to that of the rigid matrix; (2) Poisson's ratio is less than 0.5 for a rigid matrix; (3) thermal stresses in the matrix surrounding the particles reduce the apparent modulus of the polymer matrix because of the nonlinear stress—strain behavior of the matrix. This latter effect gives rise to a temperature dependence of the relative modulus below the glass transition temperature of the polymer matrix. Formation of strong aggregates increases the shear modulus the same as viscosity is increased by aggregation. Torsion or flexure tests on specimens made by casting or by molding give incorrect low values of moduli because of a surface layer containing an excess of matrix material; this gives rise to a fictitious increase in apparent modulus as particle size decreases. The mechanical damping can be markedly changed by surface treatment of the filler particles without noticeable changes in the modulus. The Kerner equation, which is a lower bound to the shear modulus, is modified and brought into closer aggrement with the experimental data by taking into account the maximum packing fraction of the filler particles.

A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes

Abstract: General piecewise linear constitutive laws with associated flow rules are formulated in matrix notation; some properties and specializations (in particular to kinematic and isotropic hardening) are discussed. With reference to finite element models of structures and, hence, in matrix-vector description, the following results are achieved: a) the holonomic solutions to the analysis problem for given loads and dislocations are shown to be characterized by means of six “quadratic-linear” minimum principles, two of general, four of conditioned validity;b) the incremental counterparts of the above theorems are indicated by analogy; some comparison properties concerning holonomic and nonholonomic solutions, are pointed out;c) a shakedown theorem is established for variable repeated loads and dislocations, with allowance for inertia forces and viscous damping, i. e. a generalization to workhardening structures of Ceradini's and (in quasi-static situations) Melan's theorems;d) a method is proposed for evaluating under holonomy hypothesis, or bounding from above, the safety factor with respect to local failure due to limited plastic strain capacity.

An Asymptotic χ2 Test for the Equality of Two Correlation Matrices

Abstract: An asymptotic χ2 test for the equality of two correlation matrices is derived. The key result is a simple representation for the inverse of the asymptotic covariance matrix of a sample correlation matrix. The test statistic has the form of a standard normal theory statistic for testing the equality of two covariance matrices with a correction term added. The applicability of asymptotic theory is demonstrated by two simulation studies and the statistic is used to test the difference in the factor patterns resulting from a set of tests given to retarded and non-retarded children. Two related tests are presented: a test for a specified correlation matrix and a test for equality of correlation matrices in two or more populations.

An Analysis of Some Graph Theoretical Cluster Techniques

Abstract: Several graph theoretic cluster techniques aimed at the automatic generation of thesauri for information retrieval systems are explored. Experimental cluster analysis is performed on a sample corpus of 2267 documents. A term-term similarity matrix is constructed for the 3950 unique terms used to index the documents. Various threshold values, T, are applied to the similarity matrix to provide a series of binary threshold matrices. The corresponding graph of each binary threshold matrix is used to obtain the term clusters.Three definitions of a cluster are analyzed: (1) the connected components of the threshold matrix; (2) the maximal complete subgraphs of the connected components of the threshold matrix; (3) clusters of the maximal complete subgraphs of the threshold matrix, as described by Gotlieb and Kumar.Algorithms are described and analyzed for obtaining each cluster type. The algorithms are designed to be useful for large document and index collections. Two algorithms have been tested that find maximal complete subgraphs. An algorithm developed by Bierstone offers a significant time improvement over one suggested by Bonner.For threshold levels T ≥ 0.6, basically the same clusters are developed regardless of the cluster definition used. In such situations one need only find the connected components of the graph to develop the clusters.

A note on pole assignment in linear systems with incomplete state feedback

Abstract: The following system is considered: \dot{x}= Ax + Bu y = Cx where x is an n vector describing the state of the system, u is an m vector of inputs to the system, and y is an l vector ( l \leq n ) of output variables. It is shown that if rank C = l , and if (A,B) are controllable, then a linear feedback of the output variables u = K*y, where K*is a constant matrix, can always be found, so that l eigenvalues of the closed-loop system matrix A + BK*C are arbitrarily close (but not necessarily equal) to l preassigned values. (The preassigned values must be chosen so that any complex numbers appearing do so in complex conjugate pairs.) This generalizes an earlier result of Wonham [1]. An algorithm is described which enables K*to be simply found, and examples of the algorithm applied to some simple systems are included.

Higher random-phase approximation as an approximation to the equations of motion.

Abstract: Starting from the equations of motion expressed as ground-state expectation values, we have derived a higher-order random-phase approximation (RPA) for excitation frequencies of low-lying states. The matrix elements in the expectation value are obtained up to terms linear in the ground-state correlation coefficients. We represent the ground state as eU|HF〉, where U is a linear combination of two particle-hole operators, and |HF〉 is the Hartree-Fock ground state. We then retain terms only up to those linear in the correlation coefficients in the equation determining the ground state. This equation and that for the excitation energy are then solved self-consistently. We do not make the quasiboson approximation in this procedure, and explicitly discuss the overcounting characteristics of this approximation. The resulting equations have the same form as those of the RPA, but this higher RPA removes many deficiencies of the RPA.

On the number of multiplications necessary to compute certain functions

Abstract: : The number of multiplications and divisions required in certain computations is investigated. In particular, results of Pan Motzkin, about polynomial evaluation as well as similar results about the product of a matrix by vector, are obtained. As an application of the results on the product of a matrix by vector, a new algorithm for matrix multiplication, which requires about 1/2(n cubed) multiplications, is obtained.

Return-difference and return-ratio matrices and their use in analysis and design of multivariable feedback control systems

Abstract: Bode's concepts of return difference and return ratio are shown to play a fundamental role in the analysis of multivariable feedback control systems. Matrix transfer functions are regarded as operators on linear vector spaces over the field of rational functions in the complex variable s. The eigenvalues of such operators are identified as characteristic transfer functions. The corresponding characteristic frequency responses provide a simple and natural link between classical single-loop design techniques and multivariable-system feedback theory. These concepts then serve as a unifying thread in a coherent and systematic discussion of multivariable-feedback-system design techniques.

Efficient Code for Steady-State Flows in Networks

Abstract: A complete algorithm for the computer solution of steady-state fluid flows in networks is given. Particular stress is placed on fast solution, minimal storage requirements and simplicity of the input data. Although the Hardy Cross method is the classical method of solution of this type of problem, convergence is slow for large networks. To overcome this problem, the whole network is considered simultaneous, and this produces a large system of non-linear equations. Newton's method is applied, which results in an iterative solution of a system of linear equations. In order to reduce computer storage requirements and to simplify the data input, a number of algorithms from graph theory are involved. The resulting matrix of coefficients associated with the system of linear equations is banded and symmetric for which efficient (in time and memory requirements) methods of solution exist.

Matrix transformations between some classes of sequences

Abstract: Let A = (ank) be an infinite matrix of complex numbers ank (n, k = 1, 2,…) and X, Y two subsets of the space s of complex sequences. We say that the matrix A defines a (matrix) transformation from X into Y, and we denote it by writing A: X → Y, if for every sequence x = (xk)∈X the sequence Ax = (An(x)) is in Y, where An(x) = Σankxk and the sum without limits is always taken from k = 1 to k = ∞. The sequence Ax is called the transformation of x by the matrix A. By (X, Y) we denote the class of matrices A such that A: X → Y.

On the design of optimal constrained dynamic compensators for linear constant systems

Abstract: The design of linear time-invariant dynamic compensators of fixed dimensionality s , which are to be used for the regulation of an n th-order linear time-invariant plant, is dealt with. A modified quadratic cost criterion is employed in which a quadratic penalty on the system state as well as all compensator gains is used; the effects of the initial state are averaged out. The optimal compensator gains are specified by a set of simultaneous nonlinear matrix algebraic equations. The numerical solution of these equations would specify the gain matrices of the dynamic compensator. The proposed method may prove useful in the design of low-order s compensators for high-order n plants that have few r outputs, so that the dimension of the compensator is less than that obtained through the use of the associated Kalman-Bucy filter n or the Luenberger observer n - r .

On the Matrix Riccati Equation

Abstract: The matrix Riccati equation appears in many optimal control and filtering problems. In this paper the Riccati equation is studied from an algebraic point of view, and the results are applied on optimal control of linear time invariant systems with quadratic loss.

On the discrete time matrix Riccati equation of optimal control

Abstract: This paper is concerned with the discrete time matrix Riccati equation. The properties established are those of minimality, convergence, uniqueness and stability. Further the convergence of the policy space approximation technique is proved. These results are analogous to those known for the continuous-time Riccati equation, but the techniques used are simpler.

Derivative operations on matrices

Abstract: The structure of a matrix derivative on a matrix valued function is defined. Matrix product and chain rules are developed which provide significant simplifications for obtaining derivatives of compound matrix structures, and some closed-form structures for Taylor's expansions of a matrix in terms of derivatives and elements of a second matrix are given.

Sparsity-Directed Decomposition for Gaussian Elimination on Matrices

Abstract: This is a concise critical survey of the theory and practice relating to the ordered Gaussian elimination on sparse systems. A new method of renumbering by clusters is developed, and its properties described. By establishing a correspondence between matrix patterns and directed graphs, a sequential binary partition is used to decompose the nodes of a graph into clusters. By appropriate ordering of the nodes within each cluster and by selecting clusters, one at a time, both optimal ordering and a useful form of matrix banding are achieved. Some results pertaining to the compatibility between optimal ordering for sparsity and the usual pivoting for numerical accuracy are included.

S Matrix for Yang-Mills and Gravitational Fields

Abstract: A method is suggested (and applied to the Yang-Mills and gravitational fields) for the construction of the generating functional ($S$ matrix) for fields possessing an invariance group. The unitarity and gauge independence of the $S$ matrix on the mass shell are seen explicitly.

Engine control systems

Abstract: An engine control system, particularly for a road vehicle, includes two transducers for producing two signals representing any two of the parameters engine speed, manifold pressure and throttle angle. These signals provide input to a matrix which gives an n-bit output determined by both input signals, where n is at least two. This output controls an engine characteristic.

An algorithm for numerical determination of the structure of a general matrix

Abstract: An algorithm, proposed by V. N. Kublanovskaya, for solving the complete eigenvalue problem of a degenerate (that is defective and/or derogatory) matrix, is studied theoretically and numerically. It uses successiveQR-factorizations to determine annihilated subspaces.

A general recursive procedure for analysis of variance

Abstract: SUMMARY A general recursive least squares procedure for the analysis of experimental designs is described. Any experimental design can be analyzed with a finite sequence of sweeps, in each of which a set of effects for a factor of the model is calculated and subtracted from the vector of observations. The effects are usually either simple means or effective means, which are ordinary means divided by an efficiency factor. The analysis for a particular design and model is characterized by a set of K efficiency factors for each factor of the model, where K is the order of balance of that factor, and by a triangular control matrix of indicators (0 or 1), in which subdiagonal zeros indicate orthogonality between pairs of factors in the model, and diagonal zeros indicate factors that are completely aliased with previous factors. The control matrix determines the minimal sweep sequence for analysis. The procedure may be implemented in an adaptive or 'learning' form, in which the information that characterizes the analysis is determined progressively from preliminary analyses of special dummy variates, each generated from an arbitrarily assigned set of effects for a factor of the model. A simple extension of the procedure produces the multistratum analysis required for stratified designs such as split plots and confounded factorials. Observations commonly arise from designed experiments, the symmetries and pattern of which implicitly affect the analysis and inference from the data, but are not usually explicitly characterized and utilized in general linear model formulations. This paper describes a simple procedure for least squares analysis of experimental designs with respect to a linear factorial model, in which the sequence of operations required is fully determined and controlled by the symmetries and pattern in the design. The analysis process consists of a sequence of sweeps of the data vector, determined by the factors of the model, the sweep being the only form of arithmetic operation required. In a sweep for a factor of the model, a set of effects for that factor is calculated and subtracted

A correspondence principle for strongly coupled states

Abstract: Heisenberg's correspondence principle for non-relativistic matrix elements has been generalized: quantal transition amplitudes between strongly coupled states are expressed as Fourier components of integrals over classical trajectories. The new theory reduces to Heisenberg's correspondence principle in particular cases. The theory is applicable whenever the change in quantum number is small compared with the initial quantum number. In these situations it is more comprehensive than both quantal first order perturbation theory and the sudden approximation. Furthermore, compared with standard quantal methods, the evaluation of amplitudes is particularly simple. The theory is worked out for one- and three-dimensional separable systems and the generalization to a system of arbitrary dimension is indicated.