Showing papers on "Matrix (mathematics) published in 1980"

Updating Quasi-Newton Matrices With Limited Storage

Abstract: We study how to use the BFGS quasi-Newton matrices to precondition minimization methods for problems where the storage is critical. We give an update formula which generates matrices using information from the last m iterations, where m is any number supplied by the user. The quasi-Newton matrix is updated at every iteration by dropping the oldest information and replacing it by the newest informa- tion. It is shown that the matrices generated have some desirable properties. The resulting algorithms are tested numerically and compared with several well- known methods. 1. Introduction. For the problem of minimizing an unconstrained function / of n variables, quasi-Newton methods are widely employed (4). They construct a se- quence of matrices which in some way approximate the hessian of /(or its inverse). These matrices are symmetric; therefore, it is necessary to have n(n + l)/2 storage locations for each one. For large dimensional problems it will not be possible to re- tain the matrices in the high speed storage of a computer, and one has to resort to other kinds of algorithms. For example, one could use the methods (Toint (15), Shanno (12)) which preserve the sparsity structure of the hessian, or conjugate gradient methods (CG) which only have to store 3 or 4 vectors. Recently, some CG algorithms have been developed which use a variable amount of storage and which do not require knowledge about the sparsity structure of the problem (2), (7), (8). A disadvantage of these methods is that after a certain number of iterations the quasi-Newton matrix is discarded, and the algorithm is restarted using an initial matrix (usually a diagonal matrix). We describe an algorithm which uses a limited amount of storage and where the quasi-Newton matrix is updated continuously. At every step the oldest information contained in the matrix is discarded and replaced by new one. In this way we hope to have a more up to date model of our function. We will concentrate on the BFGS method since it is considered to be the most efficient. We believe that similar algo- rithms cannot be developed for the other members of the Broyden 0-class (1). Let / be the function to be nnnimized, g its gradient and h its hessian. We define

A Comparison of the Super-CI and the Newton-Raphson Scheme in the Complete Active Space SCF Method

Abstract: A density matrix formulation is presented of the super-C I and Newton-Raphson methods in complete active space SCF (CASSCF) calculations. The CASSCF method is a special form of the MC-SCF method, where the C I wave function is assumed to be complete in a subset of the orbital space (the active space), leaving the remaining orbitals doubly occupied in all configurations. Explicit formulas are given for all matrix elements in the super-C I method and the first and second derivatives in the Newton-Raphson formulation. The similarities between the two methods are pointed out and the differences in the detailed formulations are discussed. Especially interesting is the fact, that while the second derivatives can be expressed in terms of first and second order density matrices, the matrix elements between the super-C I states involve also the third order density matrix in some cases.

Linear model reduction and solution of the algebraic Riccati equation by use of the sign function

Abstract: The sign function of a square matrix can be defined in terms of a contour integral or as the result of an iterated map $. Application of this function enables a matrix to be decomposed into two components whose spectra lie on opposite sides of the imaginary axis. This has application in reduction of linear systems to lower order models and in the solution of the matrix Lyapunov and algebraic Riccati equations.

The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators

Abstract: This paper presents a method for generating pseudo-random orthogonal matrices from the Haar distribution for the group of orthogonal matrices. The random matrices are expressed as products of $n - 1$ Householder transformations, which can be computed in $O(n^2 )$ time. The technique is used in an empirical study of two methods for estimating the condition number of a matrix.

Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method

Abstract: Small-angle scattering data from polydisperse systems can be evaluated under the assumption that all particles have the same shape and that the size distribution depends only on one linear size parameter R. The shape of the particles is assumed to be known a priori. The corresponding size distribution function for the number of particles Dn(R) or for the volume Dv(R) can be computed from the smeared, unsmoothed scattering data by the indirect transformation method restricting the range of definition of the D(R) functions to a finite range Rmin ≤ Rmax. Rmin may be equal to zero, Rmax is limited by the sampling theorem of the Fourier transformation. The resolution in real space is given by the distance of the knots of the B spline functions approximating the size distribution function. The propagated statistical error band in real space can be computed using the inverted matrix of the normal equations. The method gives satisfactory results even in those cases where the shape of the particles is not known exactly, and is superior to analytical methods if the termination effect is critical.

The Elimination Matrix: Some Lemmas and Applications

Abstract: Two transformation matrices are introduced, L and D, which contain zero and unit elements only. If A is an arbitrary $( n,n )$ matrix, L eliminates from vecA the supradiagonal elements of A, while D performs the inverse transformation for symmetricA. Many properties of L and D are derived, in particular in relation to Kronecker products. The usefulness of the two matrices is demonstrated in three areas of mathematical statistics and matrix algebra: maximum likelihood estimation of the multivariate normal distribution, the evaluation of Jacobians of transformations with symmetric or lower triangular matrix arguments, and the solution of matrix equations.

Integrable nonlinear Klein-Gordon equations and Toda lattices

Abstract: We present a class of nonlinear Klein-Gordon systems which are soluble by means of a scattering transform. More specifically, for eachN≧2 we present a system of (N−1) nonlinear Klein-Gordon equations, together with the correspondingN ×N matrix scattering problem which can be used to solve it. We illustrate these with some special examples. The general system is shown to be closely related to the equations of the periodic Toda lattice. We present a Backlund transformation and superposition formula for the general system.

Block-iterative methods for consistent and inconsistent linear equations

Abstract: We shall in this paper consider the problem of computing a generalized solution of a given linear system of equations. The matrix will be partitioned by blocks of rows or blocks of columns. The generalized inverses of the blocks are then used as data to Jacobi- and SOR-types of iterative schemes. It is shown that the methods based on partitioning by rows converge towards the minimum norm solution of a consistent linear system. The column methods converge towards a least squares solution of a given system. For the case with two blocks explicit expressions for the optimal values of the iteration parameters are obtained. Finally an application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.

Consequences of extending 1‐matrix energy functionals from pure–state representable to all ensemble representable 1 matrices

Abstract: Levy’s 1‐matrix energy functional (Ref. 4) is modified and extended to include all ensemble–representable l‐matrices in its domain. This constitutes both a generalization and a simplification of earlier observations by Gilbert. The generalization negates some criticisms of the Donnelly–Parr analysis of a 1‐matrix energy functional (Ref. 3) since it was assumed to be defined only for pure‐state representable 1‐matrices. Further study of this analysis suggests that the taking of arbitrary variations in the 1‐matrix at a certain point in their study may result in an invalid Euler equation for the 1‐matrix energy functional, although this does not alter the main results of that work. The redefinition of Levy’s functional to accommodate the larger domain allows direct application of Harriman’s analysis of the geometry of density matrices in finite dimensional situations. A decomposition of the equivalent of the Vee expectation value into two terms, one with explicit and another with implicit occupation number dependence, illustrates the role of N‐representability. The Vee expectation value is approximated by means of the Schwartz inequality.

Doubling algorithms for Toeplitz and related equations

Abstract: A new class of doubling or halving algorithms for solving Toeplitz and related equations is presented. For scalar n by n Toeplitz matrices, they require O(n \log^{2}n) computations, similarly to the HGCD (half-greatest-common-divisor) based algorithm of Gustavson and Yun. However, these new algorithms are based on the notions of "shift" or displacement rank 1 \leq \alpha \leq n , an index of how close a matrix is to being Toeplitz, requiring O(\alpha^{d} n \log^{2}n) operations, ( d \leq 2 ). A basic version of a doubling algorithm for such "α-Toeplitz matrices" is presented, and the applications of these results to related problems are mentioned, such as the inversion of banded-, block- and Hankel matrices.

Application of Maximum Likelihood Estimation of Persymmetric Covariance Matrices to Adaptive Processing

Abstract: The optimum weights for an adaptive processor are determined by solving a particular matrix equation. When, as is usually true in practice, the covariance matrix is unknown, a matrix estimator is required. Estimating the matrix can be computationally burden some. Methods of decreasing the computational burden by exploiting persymmetric symmetries are discussed. It is shown that the number of independent vector measurements required for the estimator can be decreased by up to a factor of two.

Analysis and extension of some recently proposed classical models for electronic degrees of freedom

Abstract: This paper analyzes various methods that have been developed recently for constructing a classical model for a finite set of quantum mechanical states (electronic states for our applications) and also shows how one of them, the spin matrix mapping method of Meyer and Miller, can be generalized in two aspects. First, it is shown how the methodology can be modified to obtain a classical model of any desired number of degrees of freedom, rather than only one degree of freedom as before. Second, it is shown how the method can be applied in the adiabatic representation, so as to be able to use directly the adiabatic potential energy surfaces and nonadiabatic coupling elements produced by a quantum chemistry calculation.

Linear stability of Poiseuille flow in a circular pipe

Abstract: Correction of an error in the matrix elements used by Salwen & Grosch (1972) has brought the results of the matrix-eigenvalue calculation of the linear stability of Hagen–Poiseuille flow into complete agreement with the numerical integration results of Lessen, Sadler & Liu (1968) for azimuthal index n = 1. The n = 0 results were unaffected by the error and the effect of the error for n > 1 is smaller than for n = 1. The new calculations confirm the conclusion that the flow is stable to infinitesimal disturbances.Further calculations have led to the discovery of a degeneracy at Reynolds number R = 61·452 ± 0·003 and wavenumber α = 0·9874 ± 0·0001, where the second and third eigenmodes have equal complex wave speeds. The variation of wave speed for these two modes has been studied in the vicinity of the degeneracy and shows similarities to the behaviour near the degeneracies found by Cotton and Salwen (see Cotton 1977) for rotating Hagen-Poiseuille flow. Finally, new results are given for n = 10 and 30; the n = 1 results are extended to R = 106; and new results are presented for the variation of the wave speed with αR at high Reynolds number. The high-R results confirm both Burridge & Drazin's (1969) slow-mode approximation and more recent fast-mode results of Burridge.

Unitary Group Approach to the Many-Electron Correlation Problem via Graphical Methods of Spin Algebras

Abstract: The various existing approaches for the evaluation of matrix elements of unitary group generators and their products with respect to the basis of electronic Gelfand states or the corresponding Yamanouchi-Kotani states are interrelated, and their desirable features combined, yielding a direct algorithm for the evaluation of matrix elements of products of two generators and, consequently, a simple and efficient algorithm for the calculation of two-electron matrix elements of spin-independent Hamiltonians needed in the unitary group configuration interaction (shell model) approach. Moreover, this algorithm is compatible with the efficient generation and representation scheme for electronic Gelfand states based on the distinct row table concept. Diagrammatic techniques based on the time-independent Wick theorem and graphical methods of spin algebras are used to derive the required factors for both one and two-generator (or electron) matrix elements for three different phase conventions and several possible simplifications in the evaluation of the two-electron part of the Hamiltonian matrix are outlined.

Sample Size Considerations for Adaptive Arrays

Abstract: Digital control of adaptive arrays has been shown to be a feasible alternative to analog feedback-loop control. As the eigenvalue spread of the correlation matrix no longer controls the speed of adaption, one merely has to ensure that enough samples have been taken so that the matrix estimate is close to the true matrix. While previous studies have assumed ideal conditions, it is shown here that if the true signal direction is not known exactly or if the data containing the interference are corrupted by a desired signal, then more samples are required to ensure that the estimated weighting vector gives a near optimal performance.

The mass spectrum and the $S$-matrix of the massive Thirring model in the repulsive case

Abstract: The repulsive case of the quantum version of the massive Thirring model is considered. It is shown that there is a rich particle spectrum in the theory. TheS matrix of fermions proves to be a discontinuous function of the coupling constant. These effects are the result of the qualitative change of the physical vacuum in the limit of the strong repulsiong →−π.

Theory of resonances in the electromagnetic scattering by macroscopic bodies

Abstract: The electromagnetic scattering resonances of a collection of macroscopic bodies with uniform electric properties are used to construct a spectral representation for the scattered field. The resonances and their weights are found by solving for the eigenvalues and eigenstates of a non-Hermitian, linear integral operator $\ensuremath{\Gamma}$. A scheme is developed for doing this by diagonalizing a matrix that represents $\ensuremath{\Gamma}$ by the set of individual grain eigenstates---the diagonal elements are individual grain eigenvalues while the off-diagonal elements are overlap integrals of eigenstates from two different grains. For a system of spherical scatterers, this scheme leads to a reasonable method of calculating numerically the scattered field in cases where the multiple scattering is important. As an example, the scattering by a pair of identical spheres is worked out analytically for a limiting case. Sum rules for the weights in the spectral representation are derived and discussed.

Infinite matrices of operators

Abstract: Notation and terminology.- Generalized Kothe-Toeplitz duals.- Characterization of matrix classes.- Tauberian theorems.- Consistency theorems.- Operator Norlund means.

Some general algorithms. I: Arithmetic groups

Abstract: An algebraic matrix group of degree n, defined over Q, is a subgroup of GLn(C) which is the set of common zeros in GLn(C) of finitely many polynomials, with rational coefficients, in the n2 matrix entries. We shall also call such a group a Q-group, of degree n. We say that the Q-group is given explicitly if these polynomials are explicitly given. If G is such a group and R is a subring of C, put

Factorizations of Transfer Functions

Abstract: This paper is concerned with minimal factorizations of rational matrix functions. The treatment is based on a new geometrical principle. In fact, it is shown that there is a one-to-one correspondence between minimal factorizations on the one hand and certain projections on the other. Considerable attention is given to the problem of stability of a minimal factorization. Also the numerical aspects are discussed. Along the way, a stability theorem for solutions of the matrix Riccati equation is obtained.

Some matrix elements for morse oscillators

Abstract: In this paper general working equations for the Morse (r-r,)' rnatrix elernents are given. These equations can be used to calculate the diagonal (m = n) matrix elements and, for the off-diagonal (m#n) elements, are simpler to use than the ones currently available in the literature. Also, in this paper a new approach is given which allows one to obtain simple formulas, in closed forrn, for the off-diagonal matrix elements. Explicit expressions are given for 1 = 1, 2, and 3.

A reparameterization of a genetic selection index to locate its sampling properties.

Abstract: A transformation is proposed of the variables used for constructing genetic selection indices, such that the reparameterized phenotypic covariance matrix is identity and the genetic covariance matrix is diagonal. These diagonal elements play the role of heritabilities of the transformed variables. The reparameterization enables sampling properties of the index weights to be easily computed and formulae are given for data from half-sib families. The index is likely to be least stable when the transformed variables of low heritability have high economic weights, and the weights will have highest sampling variance when these heritabilities are nearly equal. It is suggested that the sample roots of the determinantal equation be inspected when constructing an index in order to give some guide to its accuracy.