Showing papers on "Matrix (mathematics) published in 1986"

Qualitative data analysis: a sourcebook of new methods

Abstract: Part One: Introduction Part Two: Focusing and Bounding the Collection of Data Part Three: Analysis During Data Collection Part Four: Within-Site Analysis Part Five: Cross-Site Analysis Part Six: Matrix Displays: Some General Suggestions Part Seven: Drawing and Verifying Conclusions Part Eight: Concluding Remarks

Matrix fracture in fiber-reinforced ceramics

Abstract: A fiber-reinforced ceramic subject to tensile stress in the fiber direction can undergo extensive matrix cracking normal to the fibers, while the fibers remain intact. In this paper, the critical conditions for the onset of widespread matrix cracking are studied analytically on the basis of fracture mechanics theory. Two distinct situations concerning the fiber-matrix interface are contemplated : (i) unbonded fibers initially held in the matrix by thermal or other strain mismatches, but susceptible to frictional slip, and (ii) fibers that initially are weakly bonded to the matrix, but may be debonded by the stresses near the tip of an advancing matrix crack. The results generalize those of the Aveston-Cooper-Kelly theory for case (i). Optimal thermal strain mismatches for maximum cracking strength are studied, and theoretical results are compared with experimental data for a SiC fiber, lithium-alumina-silicate glass matrix composite.

A First Course in the Finite Element Method

Abstract: 1 INTRODUCTION Brief History Introduction to Matrix Notation Role of the Computer General Steps of the Finite Element Method Applications of the Finite Element Method Advantages of the Finite Element Method Computer Programs for the Finite Element Method 2 INTRODUCTION TO THE STIFFNESS (DISPLACEMENT) METHOD Definition of the Stiffness Matrix Derivation of the Stiffness Matrix for a Spring Element Example of a Spring Assemblage Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) Boundary Conditions Potential Energy Approach to Derive Spring Element Equations 3 DEVELOPMENT OF TRUSS EQUATIONS Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates Selecting Approximation Functions for Displacements Transformation of Vectors in Two Dimensions Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane Computation of Stress for a Bar in the x-y Plane Solution of a Plane Truss Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space Use of Symmetry in Structure Inclined, or Skewed, Supports Potential Energy Approach to Derive Bar Element Equations Comparison of Finite Element Solution to Exact Solution for Bar Galerkin's Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations Other Residual Methods and Their Application to a One-Dimensional Bar Problem Flowchart for Solutions of Three-Dimensional Truss Problems Computer Program Assisted Step-by-Step Solution for Truss Problem 4 DEVELOPMENT OF BEAM EQUATIONS Beam Stiffness Example of Assemblage of Beam Stiffness Matrices Examples of Beam Analysis Using the Direct Stiffness Method Distribution Loading Comparison of the Finite Element Solution to the Exact Solution for a Beam Beam Element with Nodal Hinge Potential Energy Approach to Derive Beam Element Equations Galerkin's Method for Deriving Beam Element Equations 5 FRAME AND GRID EQUATIONS Two-Dimensional Arbitrarily Oriented Beam Element Rigid Plane Frame Examples Inclined or Skewed Supports - Frame Element Grid Equations Beam Element Arbitrarily Oriented in Space Concept of Substructure Analysis 6 DEVELOPMENT OF THE PLANE STRESS AND STRAIN STIFFNESS EQUATIONS Basic Concepts of Plane Stress and Plane Strain Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations Treatment of Body and Surface Forces Explicit Expression for the Constant-Strain Triangle Stiffness Matrix Finite Element Solution of a Plane Stress Problem Rectangular Plane Element (Bilinear Rectangle, Q4) 7 PRACTICAL CONSIDERATIONS IN MODELING: INTERPRETING RESULTS AND EXAMPELS OF PLANE STRESS/STRAIN ANALYSIS Finite Element Modeling Equilibrium and Compatibility of Finite Element Results Convergence of Solution Interpretation of Stresses Static Condensation Flowchart for the Solution of Plane Stress-Strain Problems Computer Program Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress-Strain Problems 8 DEVELOPMENT OF THE LINEAR-STRAIN TRAINGLE EQUATIONS Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations Example of LST Stiffness Determination Comparison of Elements 9 AXISYMMETRIC ELEMENTS Derivation of the Stiffness Matrix Solution of an Axisymmetric Pressure Vessel Applications of Axisymmetric Elements 10 ISOPARAMETRIC FORMULATION Isoparametric Formulation of the Bar Element Stiffness Matrix Isoparametric Formulation of the Okabe Quadrilateral Element Stiffness Matrix Newton-Cotes and Gaussian Quadrature Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature Higher-Order Shape Functions 11 THREE-DIMENSIONAL STRESS ANALYSIS Three-Dimensional Stress and Strain Tetrahedral Element Isoparametric Formulation 12 PLATE BENDING ELEMENT Basic Concepts of Plate Bending Derivation of a Plate Bending Element Stiffness Matrix and Equations Some Plate Element Numerical Comparisons Computer Solutions for Plate Bending Problems 13 HEAT TRANSFER AND MASS TRANSPORT Derivation of the Basic Differential Equation Heat Transfer with Convection Typical Units Thermal Conductivities K and Heat-Transfer Coefficients, h One-Dimensional Finite Element Formulation Using a Variational Method Two-Dimensional Finite Element Formulation Line or Point Sources Three-Dimensional Heat Transfer by the Finite Element Method One-Dimensional Heat Transfer with Mass Transport Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method Flowchart and Examples of a Heat-Transfer Program 14 FLUID FLOW IN POROUS MEDIA AND THROUGH HYDRAULIC NETWORKS AND ELECTRICAL NETWORKS AND ELECTROSTATICS Derivation of the Basic Differential Equations One-Dimensional Finite Element Formulation Two-Dimensional Finite Element Formulation Flowchart and Example of a Fluid-Flow Program Electrical Networks Electrostatics 15 THERMAL STRESS Formulation of the Thermal Stress Problem and Examples 16 STRUCTURAL DYNAMICS AND TIME-DEPENDENT HEAT TRANSFER Dynamics of a Spring-Mass System Direct Derivation of the Bar Element Equations Numerical Integration in Time Natural Frequencies of a One-Dimensional Bar Time-Dependent One-Dimensional Bar Analysis Beam Element Mass Matrices and Natural Frequencies Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices Time-Dependent Heat-Transfer Computer Program Example Solutions for Structural Dynamics APPENDIX A - MATRIX ALGEBRA Definition of a Matrix Matrix Operations Cofactor of Adjoint Method to Determine the Inverse of a Matrix Inverse of a Matrix by Row Reduction Properties of Stiffness Matrices APPENDIX B - METHODS FOR SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS Introduction General Form of the Equations Uniqueness, Nonuniqueness, and Nonexistence of Solution Methods for Solving Linear Algebraic Equations Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods APPENDIX C - EQUATIONS FOR ELASTICITY THEORY Introduction Differential Equations of Equilibrium Strain/Displacement and Compatibility Equations Stress-Strain Relationships APPENDIX D - EQUIVALENT NODAL FORCES APPENDIX E - PRINCIPLE OF VIRTUAL WORK APPENDIX F - PROPERTIES OF STRUCTURAL STEEL AND ALUMINUM SHAPES ANSWERS TO SELECTED PROBLEMS INDEX

A three-dimensional finite-strain rod model. Part II: Computational aspects

Abstract: The variational formulation and computational aspects of a three-dimensional finite-strain rod model, considered in Part I, are presented. A particular parametrization is employed that bypasses the singularity typically associated with the use of Euler angles. As in the classical Kirchhoff-Love model, rotations have the standard interpretation of orthogonal, generally noncommutative, transformations. This is in contrast with alternative formulations proposed by Argyris et al. [5–8], based on the notion of semitangential rotation. Emphasis is placed on a geometric approach, which proves essential in the formulation of algorithms. In particular, the configuration update procedure becomes the algorithmic counterpart of the exponential map. The computational implementation relies on the formula for the exponential of a skew-symmetric matrix. Consistent linearization procedures are employed to obtain linearized weak forms of the balance equations. The geometric stiffness then becomes generally nonsymmetric as a result of the non-Euclidean character of the configuration space. However, complete symmetry is recovered at an equilibrium configuration, provided that the loading is conservative. An explicit condition for this to be the case is obtained. Numerical simulations including postbuckling behavior and nonconservative loading are also presented. Details pertaining to the implementation of the present formulation are also discussed.

Quantum $R$ matrix for the generalized Toda system

Abstract: We report the explicit form of the quantumR matrix in the fundamental representation for the generalized Toda system associated with non-exceptional affine Lie algebras.

Solving sparse linear equations over finite fields

Abstract: A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described. The algorithms discussed all require O(n_{1}(\omega + n_{1})\log^{k}n_{1}) field operations, where n_{1} is the maximum dimension of the coefficient matrix, \omega is approximately the number of field operations required to apply the matrix to a test vector, and the value of k depends on the algorithm. A probabilistic algorithm is shown to exist for finding the determinant of a square matrix. Also, probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error.

Invariant subspaces of matrices with applications

Abstract: FUNDAMENTAL PROPERTIES OF INVARIANT SUBSPACES AND APPLICATIONS: Invariant Subspaces: Definition, Examples and First Properties Jordan Form and Invariant Subspaces Coinvariant and Semi-invariant Subspaces Jordan Form for Extensions and Completions Applications to Matrix Polynomials Invariant Subspaces for Transformations between Different Spaces Rational Matrix Functions Linear Systems ALGEBRAIC PROPERTIES OF INVARIANT SUBSPACES: Commuting Matrices and Hyperinvariant Subspaces Description of Invariant Subspaces and Linear Transformations with the Same Invariant Subspaces Algebras of Matrices and Invariant Subspaces Real Linear Transformations TOPOLOGICAL PROPERTIES OF INVARIANT SUBSPACES AND STABILITY: The Metric Space of Subspaces The Metric Space of Invariant Subspaces Continuity and Stability of Invariant Subspaces Perturbations of Lattices of Invariant Subspaces with Restrictions Applications ANALYTIC PROPERTIES OF INVARIANT SUBSPACES: Analytic Families of Subspaces Jordan Form of Analytic Matrix Functions Applications Appendix List of Notations and Conventions References.

Error propagation and figures of merit for quantification by solving matrix equations

Abstract: La propagation d'erreur et les autres figures de merite sont definis pour chaque composant, si l'on considere la partie du signal orthogonal au spectre des autres composants. Application aux donnees d'absorbance d'un melange de 4 RNA

Generalized rank annihilation factor analysis

Abstract: : The method of rank annihilation is shown to be a particular case of a more general method for quantitation in bilinear data arrays such as LC/UV, GC/ MS or emission-excitation fluorescence Generalized rank annihilation is introduced as a calibration method that allows for simultaneous quantitative determination of all the analyses of interest in a mixture of unknowns Only one calibration mixture is required The bilinear spectra of both unknown and calibration sample must be obtained Bilinear target factor analysis is introduced as a projection of a target bilinear matrix onto another principal component bilinear matrix space Keywords: Multivariate analysis; Principal component regression (PCR); Two-dimensional data; Singular value decomposition; and Pseudoinverse

Computing the polar decomposition with applications

Abstract: A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate of convergence and it is shown how reliable estimates of the optimal parameters may be computed in practice.To add to the known best approximation property of the unitary polar factor, the Hermitian polar factor H of a nonsingular Hermitian matrix A is shown to be a good positive definite approximation to Aand $\frac{1}{2}(A + H)$ is shown to be a best Hermitian positive semi-definite approximation to A. Perturbation bounds for the polar factors are derived.Applications of the polar decomposition to factor analysis, aerospace computations and optimisation are outlined; and a new method is derived for computing the square root of a symmetric positive definite matrix.

Special matrices and their applications in numerical mathematics

Abstract: 1. Basic Concepts of the Theory of Matrices.- Matrices.- Determinants.- Nonsingular matrices. Inverse matrices.- Schur complement. Factorization.- Vector spaces. Rank.- Eigenvectors, eigenvalues. Characteristic polynomial.- Similarity. Jordan normal form.- Exercises.- 2. Symmetric Matrices. Positive Definite and Semidefinite Matrices.- Euclidean and unitary spaces.- Symmetric and Hermitian matrices.- Orthogonal and unitary matrices.- Gram-Schmidt orthonormalization. Schur's theorem.- Positive definite and positive semidefinite matrices.- Sylvester's law of inertia.- Singular value decomposition.- Exercises.- 3. Graphs and Matrices.- Digraphs.- Digraph of a matrix.- Undirected graphs. Trees.- Bigraphs.- Exercises.- 4. Nonnegative Matrices. Stochastic and Doubly Stochastic Matrices.- Nonnegative matrices.- The Perron-Frobenius theorem.- Cyclic matrices.- Stochastic matrices.- Doubly stochastic matrices.- Exercises.- 5. M-Matrices (Matrices of Classes K and K0).- Class K.- Class K0.- Diagonally dominant matrices.- Monotone matrices.- Class P.- Exercises.- 6. Tensor Product of Matrices. Compound Matrices.- Tensor product.- Compound matrices.- Exercises.- 7. Matrices and Polynomials. Stable matrices.- Characteristic polynomial.- Matrices associated with polynomials.- Bezout matrices.- Hankel matrices.- Toeplitz and Lowner matrices.- Stable matrices.- Exercises.- 8. Band Matrices.- Band matrices and graphs.- Eigenvalues and eigenvectors of tridiagonal matrices.- Exercises.- 9. Norms and Their Use for Estimation of Eigenvalues.- Norms.- Measure of nonsingularity. Dual norms.- Bounds for eigenvalues.- Exercises.- 10. Direct Methods for Solving Linear Systems.- Nonsingular case.- General case.- Exercises.- 11. Iterative Methods for Solving Linear Systems.- The Jacobi method.- The Gauss-Seidel method.- The SOR method.- Exercises.- 12. Matrix Inversion.- Inversion of special matrices.- The pseudoinverse.- Exercises.- 13. Numerical Methods for Computing Eigenvalues of Matrices.- Computation of selected eigenvalues.- Computation of all the eigenvalues.- Exercises.- 14. Sparse matrices.- Storing. Elimination ordering.- Envelopes. Profile.- Exercises.

Matrix formulations for the transfer of solar radiation in a plane-parallel scattering atmosphere.

Abstract: Matrix formulations are presented for the discrete-ordinate and the matrix-operator methods of solving the transfer of solar radiation in a plane-parallel scattering atmosphere. Eigenspace transformations of symmetric matrices are introduced into the method of Stamnes and Swanson instead of using the decomposition of an asymmetric matrix. The computational stability is considerably improved by this algorithm, especially for single-precision calculations. Representations of the reflection and transmission matrices in the matrix-operator method are also given, in terms of the indicated formulations, by considering a boundary-value problem of the discrete-ordinate method. The solutions of the discrete-ordinate method for inhomogeneous atmospheres are given by combining discrete-ordinate solutions for respective homogeneous sublayers through the addition technique of the matrix-operator method.

On distributed Gaussian bases for simple model multidimensional vibrational problems

Abstract: Distributed Gaussian bases (DGB) are defined and used to calculate eigenvalues for one and multidimensional potentials. Comparisons are made with calculations using other bases. The DGB is shown to be accurate, flexible, and efficient. In addition, the localized nature of the basis requires only very low order numerical quadrature for the evaluation of potential matrix elements.

Solution of the matrix equation AX - XB = C

Abstract: In this paper a finite algebraic formula for the solution of the matrix equationAX−XB=C is derived. Based on it, a new direct algorithm is given.

Fault-tolerant matrix arithmetic and signal processing on highly concurrent computing structures

Abstract: Hardware for executing matrix arithmetic and signal processing algorithms at high speeds is in great demand in many real-time and scientific applications. With the advent of VLSI technology, large numbers of processing elements which cooperate with each other at high speed have become economically feasible. Since any functional error in a high-performance system may seriously jeopardize the operation of the system and its data integrity, some level of fault tolerance must be incorporated in order to ensure that the results of long computations are valid. Since the major computational requirements for many important real-time signal processing tasks can be reduced to a common set of basic matrix operations, the development of a unified fault-tolerant scheme for matrix operations can solve the problems of both reliable signal processing and reliable matrix operations. Earlier work proposed a low-cost checksum scheme for fault-tolerant matrix operations on multiple processor systems. However, this scheme can only correct errors in matrix multiplication; it can detect, but not correct, errors in matrix-vector multiplication, LU decomposition, matrix inversion, etc. In order to solve these problems with the checksum scheme, a very general matrix encoding scheme is proposed in this paper to achieve fault-tolerant matrix arithmetic and signal processing with linear arrays, which are believed to hold the most promise in VLSI computing structures for their flexibility, low cost, and applicability to most of the interesting algorithms. This proposed technique is, therefore, a very cost-effective encoding technique to achieve fault-tolerant matrix arithmetic and signal processing on highly concurrent VLSI computing structures.

Feedback stabilization of linear autonomous time lag systems

Abstract: A stabilization theory which employs well-established finite-dimensional control system tools is developed for the stabilization of linear autonomous time lag systems. The main ideas include 1) a set whose elements are matrices each of which is a left zero of the system characteristic quasi-polynomial matrix, and 2) a linear transformation which reduces the delay system to a delay-free system whose state matrix is a direct sum of N elements of the matrix set where N is some positive integer. From the definition of this matrix set, it is shown that each of its elements inherits its spectrum from that of the delay system so that by design, the system unstable poles may be embedded in the spectrum of the delay-free system. Under the assumption of spectral stabilizability, it is then shown how to obtain a stabilizing feedback control law on the basis of the delay-free system. Numerical examples are presented to confirm the theory.

Matrix Theory: A Second Course

Abstract: 1. Review of Basic Background.- 2. Linear Spaces and Operators.- 3. Canonical Forms.- 4. Quadratic Forms and Optimization.- 5. Differential and Difference Equations.- 6. Other Topics.- References.

A survey of matrix inverse eigenvalue problems

Abstract: In this paper, we present a survey of some recent results regarding direct methods for solving certain symmetric inverse eigenvalue problems. The problems we discuss in this paper are those of generating a symmetric matrix, either Jacobi, banded, or some variation thereof, given only some information on the eigenvalues of the matrix itself and some of its principal submatrices. Much of the motivation for the problems discussed in this paper came about from an interest in the inverse Sturm-Liouville problem. A preliminary version of this report was issued as a technical report of the Computer Science Department, University of Minnesota, TR 86-20, May 1986.

Stability and the matrix Lyapunov equation for discrete 2-dimensional systems

Abstract: The stability of two-dimensional, linear, discrete systems is examined using the 2-D matrix Lyapunov equation. While the existence of a positive definite solution pair to the 2-D Lyapunov equation is sufficient for stability, the paper proves that such existence is not necessary for stability, disproving a long-standing conjecture.

Comparison of finite element techniques for solidification problems

Abstract: The accuracies of the computed temperatures of a liquid in a corner region under freezing conditions are compared for various fixed-grid finite element techniques using the analytical solution for this problem as a reference. In the finite element formulation of the problem different time-stepping schemes are compared: the implicit Euler-backward algorithm combined with an iterative scheme and two three-time-level methods—the Lees algorithm and a Dupont algorithm, which are both applied as non-iterative schemes. Furthermore, different methods for handling the evolution of latent heat are examined: an approximation method suggested by Lemmon and one suggested by Del Giudice, both using the enthalpy formulation as well as a fictitious heat-flow method presented by Rolph and Bathe. Results of calculations performed with the consistent heat-capacity matrix are compared with those performed with a lumped heat-capacity matrix.

Distribution of matrix elements of chaotic systems

Abstract: When a quantum system has a chaotic classical analog, its matrix elements in the energy representation are closely related to various microcanonical averages of the classical system. The diagonal matrix elements cluster around the classical expectation values, with fluctuations similar to the values of the off-diagonal matrix elements. The latter in turn are related to the classical autocorrelations. These results imply that quantum perturbation theory must fail, for chaotic systems, in the semiclassical limit \ensuremath{\Elzxh}\ensuremath{\rightarrow}0: Two arbitrarily close Hamiltonians have, in general, completely different sets of eigenvectors.

Endothelial cell-matrix interactions: in vitro models of angiogenesis.

Abstract: Introduction The endothelium is composed of a number of heterogeneous cell populations from a variety of vascular beds. Endothelial cells from these diverse beds share some common structures and functions but also exhibit a wide range of diversity in their morphological appearance, function, and response to injury (2,6,22). One factor that is thought to play key, if not pivotal, roles in the modulation of endothelial cell behavior is the extracellular matrix (21,24,2 5). A great deal of morphological evidence (at light and electron microscopic levels) supports the notion that the nature of the subendothelial matrix underlying endothelial cells varies depending on such criteria as size and type of artery and the particular microvascular bed examined (15,36,37). These differences in matrix, although still incompletely documented biochemically and immunochemicaliy, have led to the hypothesis that the subendothehal matrix is a determining factor in modulating the behavior of the overlying endothelial cell population. In this review we will discuss data, accrued from in vitro systems, that support this notion. Both large vessel (arterial) and microvascular (capillary) endothelial cell culture systems will be discussed. Because of the limited length of this review, its content is selective and will focus on the data of only several of the many laboratories actively working in this area.

Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation

Abstract: Lower and upper bounds on the trace of the positive semidefinite solution of the algebraic matrix Riccati and Lyapunov equation are derived. The upper trace bound obtained in this note in many cases results in a tighter bound as compared to the Upper bound for the maximal eigenvalue proposed in [1] and [2].

Liquid crystal matrix display having improved spacers and method of making same

Abstract: Light influencing displays and more particularly liquid crystal displays are disclosed which have a pair of light transmissive electrode supports which are uniformly spaced apart by a predetermined distance over the entire display area. The electrode supports are spaced apart by spacer means including a plurality of spacers arranged in a predetermined pattern between the electrode supports.

On the eigenvalue distribution of a class of preconditioning methods

Abstract: A class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationAx=b, whereA is a symmetric positive definite matrix. The methods are based on an incomplete factorization of the matrixA and include both pointwise and blockwise factorization. We study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofC−1A, whereC is the preconditioning matrix. We also show graphic representations of the eigenvalues and present numerical tests of the methods.