Showing papers on "Matrix (mathematics) published in 1987"

Least-Squares Fitting of Two 3-D Point Sets

Abstract: Two point sets {pi} and {p'i}; i = 1, 2,..., N are related by p'i = Rpi + T + Ni, where R is a rotation matrix, T a translation vector, and Ni a noise vector. Given {pi} and {p'i}, we present an algorithm for finding the least-squares solution of R and T, which is based on the singular value decomposition (SVD) of a 3 × 3 matrix. This new algorithm is compared to two earlier algorithms with respect to computer time requirements.

Substitution dynamical systems, spectral analysis

Abstract: The Banach Algebra (T).- Spectral Theory of Unitary Operators.- Spectral Theory of Dynamical Systems.- Dynamical Systems Associated with Sequences.- Dynamical Systems Arising from Substitutions.- Eigenvalues of Substitution Dynamical Systems.- Matrices of Measures.- Matrix Riesz Products.- Bijective Automata.- Maximal Spectral Type of General Automata.- Spectral Multiplicity of General Automata.- Compact Automata.

Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms

Abstract: An algorithm is presented in this paper for computing state-space balancing transformations directly from a state-space realization. The algorithm requires no "squaring up" or unnecessary matrix products. Various algorithmic aspects are discussed in detail. A key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product. Other contragredient transformation applications are also described. It is further shown that a similar approach may be taken, involving the generalized singular value decomposition, to the classical simultaneous diagonalization problem. These SVD-based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems.

The generalized group technology concept

Abstract: In this paper two classes of clustering models are considered: (1) matrix, and (2) integer programming. The relationship between the matrix model, the p-median model and the classical group technology concept is discussed. A generalized group technology concept, based on generation for one part of a number of different process plans, is proposed. This new concept improves the quality of process (part) families and machine cells. A corresponding integer programming model is formulated. The models discussed are illustrated with numerical examples.

Optimal Orientation Detection of Linear Symmetry

Abstract: The problem of optimal detection of orientation in arbitrary neighborhoods is solved in the least squares sense. It is shown that this corresponds to fitting an axis in the Fourier domain of the n-dimensional neighborhood, the solution of which is a well known solution of a matrix eigenvalue problem. The eigenvalues are the variance or inertia with respect to the axes given by their respective eigen vectors. The orientation is taken as the axis given by the least eigenvalue. Moreover it is shown that the necessary computations can be pursued in the spatial domain without doing a Fourier transformation. An implementation for 2-D is presented. Two certainty measures are given corresponding to the orientation estimate. These are the relative or the absolute distances between the two eigenvalues, revealing whether the fitted axis is much better than an axis orthogonal to it. The result of the implementation is verified by experiments which confirm an accurate orientation estimation and reliable certainty measure in the presence of additive noise at high level as well as low levels.

Geometric applications of a matrix-searching algorithm

Abstract: LetA be a matrix with real entries and letj(i) be the index of the leftmost column containing the maximum value in rowi ofA.A is said to bemonotone ifi 1 >i 2 implies thatj(i 1) ≥J(i 2).A istotally monotone if all of its submatrices are monotone. We show that finding the maximum entry in each row of an arbitraryn xm monotone matrix requires Θ(m logn) time, whereas if the matrix is totally monotone the time is Θ(m) whenm≥n and is Θ(m(1 + log(n/m))) whenm

Multivariate Analysis of Ecological Communities

Abstract: 1 Ecological data.- 1.1 Types of data.- 1.2 Forms of data.- 1.3 Standardization and transformation of data.- 1.4 Constructing association data.- 2 Preliminary inspection of data.- 2.1 Displaying data values.- 2.2 Mapping.- 2.3 Displaying distributions of variables.- 2.4 Bivariate and multivariate displays.- 3 Ordination.- 3.1 Direct gradient analysis.- 3.2 Principal components analysis.- 3.3 Correspondence analysis.- 3.4 Ordination methods when rows or columns are grouped.- 3.5 Principal coordinates analysis.- 3.6 The horseshoe effect.- 3.7 Non-metric ordination.- 3.8 Case studies.- 4 Methods for comparing ordinations.- 4.1 Procrustes rotation.- 4.2 Generalized Procrustes analysis.- 4.3 Comparing ordination methods by multiple Procrustes analysis.- 5 Classification.- 5.1 Agglomerative hierarchical methods.- 5.2 Divisive hierarchical methods.- 5.3 Non-hierarchical classification.- 5.4 Visual displays for classification.- 5.5 Case study.- 5.6 Methods for comparing classifications.- 6 Analysis of asymmetry.- 6.1 Row and column plots.- 6.2 Skew-symmetry analysis.- 6.3 Case studies.- 6.4 A proof of the triangle-area theorem.- 7 Computing.- 7.1 Computing options.- 7.2 Examples of Genstat programs.- 7.3 Handling missing values.- 7.4 Conclusion.- 7.5 List of software.- References.- Appendix Matrix algebra.- A.1 Matrices and vectors.- A.2 Particular forms of matrices.- A.3 Simple matrix operations.- A.4 Simple geometry and some special matrices.- A.5 Matrix inversion.- A.6 Scalar functions of matrices.- A.7 Orthogonal matrices.- A.8 Matrix decompositions.- A.9 Conclusion.

A passive localization algorithm and its accuracy analysis

Abstract: The problem of estimating source location from noisy measurements of range differences (RD's) is considered. A localization technique based on solving a set of linear equations is presented and its accuracy properties are analyzed. An optimal weighting matrix for the least squares estimator is derived. The analytical expressions for the variance and bias of the estimator are validated by Monte-Carlo simulation. The problem of estimating source velocity given measurements of range differences and range-rate differences is briefly considered, and a linear equation technique is derived.

A stable finite element solution for two-dimensional magnetotelluric modelling

Abstract: Summary. We report herein on a finite element algorithm for 2-D magnetotelluric modelling which solves directly for secondary variations in the field parallel to strike, plus the subsequent vertical and transverse auxiliary fields, for both transverse electric and transverse magnetic modes. The governing Helmholtz equations for the secondary fields along strike are the same as those for total field algorithms with the addition of source terms involving the primary fields and the conductivity difference between the body and the host. Our approach has overcome a difficulty with numerical accuracy at low frequencies observed in total field solutions with 32-bit arithmetic far the transverse magnetic mode especially, but also for the transverse electric mode. Matrix ill-conditioning, which affects total field solutions, increases with the number of element rows with the square of the maximum element aspect ratio and with the inverse of frequency. In the secondary formulation, the field along strike and the auxiliary fields do not need to be extracted in the face of an approximately computed primary field which increasingly dominates the total field solution towards low frequencies. In addition to low-frequency stability, the absolute accuracy of our algorithm is verified by comparison with the TM and the TE mode analytic responses of a segmented overburden model.

Saltwater intrusion in aquifers: Development and testing of a three-dimensional finite element model

Abstract: A three-dimensional finite element model is developed for the simulation of saltwater intrusion in single and multiple coastal aquifer systems with either a confined or phreatic top aquifer. The model formulation is based on two governing equations, one for fluid flow and the other for salt transport. Density coupling of these equations is accounted for and handled using a Picard sequential solution algorithm with special provisions to enhance convergence of the iterative solution. Flexibility in the formulation allows for either three-dimensional simulations or quasi three-dimensional simulations, where flow and transport in aquitards are treated using one-dimensional analytical and/or numerical approximations. Spatial discretization of three-dimensional regions is performed using a vertical slicing approach designed to accommodate complex geometry with irregular boundaries, layering, and/or lateral discontinuity. This approach is effectively combined with the use of simple linear elements such as rectangular and triangular prisms, and composite hexahedra and pentahedra made up of tetrahedra. For these elements, computation of element matrices can be performed efficiently using influence coefficient formulas that avoid numerical integration. New transport influence coefficient formulas are presented for rectangular and triangular prism elements. Matrix assembly is performed slice by slice, and the matrix solution is achieved using a slice successive relaxation scheme. This permits a fairly large number of nodal unknowns (of the order of five to ten thousand) to be handled conveniently on small or medium-size minicomputers. Flexibility of the formulation and matrix handling procedures also allows two-dimensional and axisymmetric problems to be solved efficiently using single slice representations. Four examples are presented to demonstrate the model verification and utility. These problems represent a fair range of physical conditions. Where possible, simulation results are compared with previously published solutions.

A maximum likelihood model for estimating origin-destination matrices

Abstract: We describe a model for estimating an origin-destination matrix from an observed sample matrix, when the volumes on a subset of the links of the network and/or the total productions and attractions of the zones are known. The elements of the observed sample matrix are assumed to be integers that are obtained from independent Poisson distributions with unknown means. A maximum likelihood model is formulated to estimate these means, yielding an estimation of the “true” origin- destination matrix which is consistent with the observed link volumes. Conditions for existence and uniqueness of a solution are discussed. A solution algorithm based on the cyclic coordinate descent method is developed and its convergence properties are analyzed. The special case of the matrix estimation problem, in which marginal totals are given instead of link volumes, is considered separately; a numerical example is used to illustrate the problem. Using results about the asymptotic behavior of the distribution of the likelihood function, tests may be derived that allow statistical inferences on the consistency of the available data. Finally, an extension of the model is studied in which the observed volumes are Poisson-distributed as well.

Numerical analysis of planar optical waveguides using matrix approach

Abstract: We present here a simple matrix method for obtaining propagation characteristics, including losses for various modes of an arbitrarily graded planar waveguide structure which may have media of complex refractive indices. We show the applicability of the method for obtaining leakage losses and absorption losses, as well as for calculating beat length in directional couplers. The method involves straightforward 2 × 2 matrix multiplications, and does not require the solutions of any transcendental or differential equations.

Optical beam wave propagation through complex optical systems

Abstract: 20We describe a novel formulation of light beam propagation through any complex optical system that can be described by an ABCD ray-transfer matrix. Within the paraxial approximation, optical propagation can be formulated in terms of a Huygens principle expressed in terms of the ray-transfer ABCD matrix elements of the optical system. We extend and generalize previous treatments to include the effects of finite-sized limiting apertures (i.e., diffractive screens) in the optical train, tilt and random jitter of the optical elements, and distributed random inhomogeneities along the optical path (e.g., clear air turbulence and aerosols). In the presence of limiting apertures the ABCD matrix elements of the optical system are complex. For the case of laser beam propagation and Gaussian-shaped limiting apertures in the optical train, we obtain analytical expressions for both the spot radius and the wave-front radius of curvature at an arbitrary observation plane and give illustrative examples of practical concern. In particular, analytical expressions for the fringe visibility obtained in a coherent laser interferometric system are presented. An analytical expression for the mean spot radius of a laser beam propagating through an optical system in the presence of tilt and random jitter is obtained. We also consider the propagation of partially coherent light through optical systems. In particular, we derive a generalized van Cittert–Zernike theorem that is valid for an arbitrary optical system that can be characterized by an ABCD ray-transfer matrix. Finally, the propagation of laser beams through a general optical system in the presence of distributed random inhomogeneities is considered. An explicit expression for the mean irradiance distribution of a Gaussian-shaped beam is derived that is valid for an arbitrary optical system. In addition, we derive an expression for the mutual-coherence function for wave propagation through an arbitrary optical system. In all cases the results are expressed in terms of the ABCD matrix elements of the complete optical system. The formulation of optical propagation presented here is a rather simple and straightforward way of determining the effects of finite-sized optical elements, tilt and random jitter, and distributed random inhomogeneities along the optical path. It is merely necessary first to multiply the relevant ray matrices together to find the complete system matrix and then to substitute this matrix into the expressions given in this paper.

Truncated balanced realization of a stable non-minimal state-space system

Abstract: In this paper we present a numerically reliable algorithm to compute the balanced realization of a stable state-space system that may be arbitrarily close to being unobservable and/or uncontrollable. The resulting realization, which is known to be a good approximation of the original system, must be minimal and therefore may contain a reduced number of states. Depending on the choice of partitioning of the Hankel singular values, this algorithm can be used either as a form of minimal realization or of model reduction. This illustrates that in finite precision arithmetic these two procedures are closely related. In addition to real matrix multiplication, the algorithm only requires the solution of two Lyapunov equations and one singular value decomposition of an upper-triangular matrix.

Matrix Management: Contradictions and Insights:

Abstract: Does matrix management stifle or foster the development of new products? The arguments in favor of and against matrix consist primarily of anecdotal success or failure stories. The issue is further obscured by the failure to recognize that there are different types of matrix. Data on the usage and effectiveness of three matrix structures (functional, balanced, and project matrixes) were collected from 500 managers experienced in product development. The results indicate that matrix is still the dominant approach for completing development projects. However, while all three types of matrix have comparable usage rates, the project matrix is considered the most effective. Companies using matrix management should consider project matrix if they are trying to improve performance.

Matrix algorithms on a hypercube I: Matrix multiplication☆

Abstract: We discuss algorithms for matrix multiplication on a concurrent processor containing a two-dimensional mesh or richer topology. We present detailed performance measurements on hypercubes with 4, 16, and 64 nodes, and analyze them in terms of communication overhead and load balancing. We show that the decomposition into square subblocks is optimal C code implementing the algorithms is available.

Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems

Abstract: It is shown that solutions of linear inequalities, linear programs and certain linear complementarity problems (e.g. those with P-matrices or Z-matrices but not semidefinite matrices) are Lipschitz continuous with respect to changes in the right-hand side data of the problem. Solutions of linear programs are not Lipschitz continuous with respect to the coefficients of the objective function. The Lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right-hand side perturbation.

Normal forms of stiffness and compliance matrices

Abstract: A generalized spring associates potential energy with each position and orientation of a rigid body. The stiffness of such a spring can be represented by a 6 × 6 symmetric matrix. This matrix can be brought to a normal form by a particular choice of the coordinate frame. Analogous but independent results hold for compliance matrices. These results, obtained by using a Lie group approach, also extend the concept of the remote center of stiffness to generic generalized springs.

Nonlinear Equations and Operator Algebras

Abstract: 1. General Scheme.- 1 Generalized Derivation and Logarithmic Derivatives.- 2 Examples of Nonlinear Equations.- 3 Projection Operation.- 2. Realization of the General Scheme in Matrix Rings and N-Soliton Solutions.- 1 Wronsky Matrices.- 2 Conditions of Invertibility of Some Wronsky Matrices.- 3 N-Soliton Solutions of Nonlinear Equations.- 4 Singular Solutions of Nonlinear Equations.- 3. Realization of the General Scheme in Operator Algebras.- 1 Extenstion of Algebra C?(B(H0).- 2 Solving Linear Equations in Algebra C?(B(H)).- 3 Additional Equations.- 4 Choice of Parameters.- 5 Properties of Logarithmic Derivatives with Respect to Conjunction Operation.- 6 Invertibility Conditions for Operators $$ \hat{\Gamma } $$.- 4. Classes of Solutions to Nonlinear Equations.- 1 Examples of Solutions to Nonlinear Equations.- 2 Connection with Inverse Problems of Spectral Analysis.- 3 KP Equation.- References.

Modeling and calibration of a structured light scanner for 3-D robot vision

Abstract: In this report we have used projectivity theory to model the process of structured light scanning for 3D robot vision. The projectivity formalism is used to derive a 4 × 3 transformation matrix that converts points in the image plane into their corresponding 3D world coordinates. Calibration of the scanner consists of computing the coefficient of this matrix by showing to the system a set of lines generated by suitable object edges. We end this paper by showing how the matrix can be used to convert image pixel locations into the world coordinates of the corresponding object points using two different scanning strategies.