Showing papers on "Matrix (mathematics) published in 1985"
Book•
[...]
TL;DR: In this article, the authors present results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrate their importance in a variety of applications, such as linear algebra and matrix theory.
Abstract: Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.
23,986 citations
TL;DR: Numerical methods are described for determining robust, or well-conditioned, solutions to the problem of pole assignment by state feedback such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized.
Abstract: Numerical methods are described for determining robust, or well-conditioned, solutions to the problem of pole assignment by state feedback. The solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized. It is shown that for these solutions, upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized. A measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed-loop poles, and hence the suitability of the given poles for assignment.
1,035 citations
TL;DR: In this paper, the authors evaluated the stress for matrix cracking in brittle-matrix fiber composites using a stress intensity approach, in which the influence of the fibers that bridge the matrix crack is represented by closure tractions at the crack surfaces.
986 citations
Book•
01 Jan 1985
TL;DR: A survey of Scalar Polynomials can be found in this article, where the Jordan Canonical Form is used to define the normal form of matrix polynomials and normal forms.
Abstract: Maxtrix Algebra. Determinants, Inverse Matrices, and Rank. Linear, Euclidean, and Unitary Spaces. Linear Transformations and Matrices. Linear Transformations in Unitary Spaces and Simple Matrices. The Jordan Canonical Form: A Geometric Approach. Matrix Polynomials and Normal Forms. The Variational Method. Functions of Matrices. Norms and Bounds for Eigenvalues. Perturbation Theory. Linear Matrix Equations and Generalized Inverses. Stability Problems. Matrix Polynomials. Nonnegative Matrices. Appendix 1. A Survey of Scalar Polynomials. Appendix 2. Some Theorems and Notions from Analysis. Appendix 3. Suggestions for Further Reading. Index.
748 citations
TL;DR: In this article, a realizability-based approach to compute the effective moduli of composites is presented. The method is based on the idea that the composite is constructed explicitly from an initial material through a series of incremental additions, and the construction process is uniquely specified by parametrizing the volume fractions of the included phases.
688 citations
TL;DR: The basic algorithm for simulating wave propagation in two- and three-dimensional transmission-live networks is derived and the introduction of boundaries, dielectric and magnetic materials, losses, and anisotropy are discussed in detail.
Abstract: This paper presents an overview of the transmission-line matrix (TLM) method of analysis, describing its historical background from Huygens's principle to modem computer formulations. The basic algorithm for simulating wave propagation in two- and three-dimensional transmission-live networks is derived. The introduction of boundaries, dielectric and magnetic materials, losses, and anisotropy are discussed in detail. Furthermore, the various sources of error and the limitations of the method are given, and methods for error correction or reduction, as well as improvements of numerical efficiency, are discussed. Finally, some typical applications to microwave problems are presented.
678 citations
TL;DR: In this article, the Table Cl method for electronic structure calculations is presented, with emphasis on the computation of matrix elements between configurations possessing equal numbers of open shells (ΔK = 0).
Abstract: Organizational details for the Table Cl method for electronic structure calculations are presented, with emphasis on the computation of matrix elements between configurations possessing equal numbers of open shells (ΔK = 0). The calculations are divided into five distinct cases based on the type of electron-repulsion integrals required to evaluate the various types of matrix elements. A moderately-sized table of small integers is constructed which allows the avoidance of all comparison of determinantal basis functions, without sacrificing the utility of a configuration-driven approach in configuration-selection methods, which have demonstrated a high degree of applicability in the calculation of ground and excited electronic states of atoms and molecules.
546 citations
TL;DR: In this article, a generalized inverse (HTWH)-1HTW solution was proposed, where W is a positive definite weighting matrix (usually taken to be the inverse of the measurement covariance matrix).
Abstract: The global positioning system (GPS) equations are usually solved with an application of Newton's method or a variant thereof: Xn+1 = xn + H-1(t - f(xn)). (1) Here x is a vector comprising the user position coordinates together with clock offset, t is a vector of tour pseudorange measurements, and H is a measurement matrix of partial derivatives H = fx· In fact the first fix of a Kalman filter provides a solution of this type. If more than four pseudoranges are available for extended batch processing, H-1 may be replaced by a generalized inverse (HTWH)-1HTW, where W is a positive definite weighting matrix (usually taken to be the inverse of the measurement covariance matrix). This paper introduces a new method of solution that is algebraic and noniterative in nature, computationally efficient and numerically stable, admits extended batch processing, improves accuracy in bad geometric dilution of precision (GDOP) situations, and allows a "cold start" in deep space applications.
417 citations
TL;DR: In this paper, the complete characterization of scattered light is described in the context of Stokes vectors and Mueller matrices which highly motivates the measuring procedures, and the most general form of the scattering matrix coupled with polarizers and quarter wave plates elegantly demonstrates the physical relationship among the matrix elements and polarization measurements.
Abstract: The complete characterization of scattered light is described in the context of Stokes vectors and Mueller matrices which highly motivates the measuring procedures. The most general form of the scattering matrix coupled with polarizers and quarter wave plates elegantly demonstrates the physical relationship among the matrix elements and polarization measurements.
383 citations
TL;DR: Two classes of matrix splittings are presented and applications to the parallel iterative solution of systems of linear equations are given, resulting in algorithms which can be implemented efficiently on parallel computing systems.
Abstract: We present two classes of matrix splittings and give applications to the parallel iterative solution of systems of linear equations. These splittings generalize regular splittings and P-regular splittings, resulting in algorithms which can be implemented efficiently on parallel computing systems. Convergence is established, rate of convergence is discussed, and numerical examples are given.
353 citations
TL;DR: Numerical experiments on test problems for two dimensions indicate that a particularly attractive preconditioning, which uses special properties of tridiagonal matrix inverses, can be computationally more efficient for the same computer storage than other preconditionsings, including the popular point incomplete Cholesky factorization.
Abstract: Block preconditionings for the conjugate gradient method are investigated for solving positive definite block tridiagonal systems of linear equations arising from discretization of boundary value problems for elliptic partial differential equations. The preconditionings rest on the use of sparse approximate matrix inverses to generate incomplete block Cholesky factorizations. Carrying out of the factorizations can be guaranteed under suitable conditions. Numerical experiments on test problems for two dimensions indicate that a particularly attractive preconditioning, which uses special properties of tridiagonal matrix inverses, can be computationally more efficient for the same computer storage than other preconditionings, including the popular point incomplete Cholesky factorization.
TL;DR: The structure and implementation of a new general iterative method for diagonalising large matrices (the 'residual minimisation/direct inversion in the iterative subspace' method of Bendt and Zunger) are described and contrasted with other more commonly used iterative techniques.
Abstract: The structure and implementation of a new general iterative method for diagonalising large matrices (the 'residual minimisation/direct inversion in the iterative subspace' method of Bendt and Zunger) are described and contrasted with other more commonly used iterative techniques. The method requires the direct diagonalisation of only a small submatrix, does not require the storage of the large matrix and provides eigensolutions to within a prescribed precision in a rapidly convergent iterative procedure. Numerical results for two rather different matrices (a real 50*50 non-diagonally dominant matrix and a complex Hermitian 181*181 matrix corresponding to the pseudopotential band structure of a semiconductor in a plane wave basis set) are used to compare the new method with the competing methods. the new method converges quickly and should be the most efficient for very large matrices in terms both of computation time and central storage requirements; it is quite insensitive to the properties of the matrices used. This technique makes possible efficient solution of a variety of quantum mechanical matrix problems where large basis set expansions are required.
TL;DR: Parallel Jacobi-like algorithms are presented for computing a singular-value decomposition of an $(m \geqq n) matrix and an eigenvalue decompositions of an $n \times n$ symmetric matrix.
Abstract: Parallel Jacobi-like algorithms are presented for computing a singular-value decomposition of an $m \times n$ matrix $(m \geqq n)$ and an eigenvalue decomposition of an $n \times n$ symmetric matrix. A linear array of $O(n)$ processors is proposed for the singular-value problem; the associated algorithm requires time $O(mnS)$, where S is the number of sweeps (typically $S \leqq 10$). A square array of $O(n^2 )$ processors with nearest-neighbor communication is proposed for the eigenvalue problem; the associated algorithm requires time $O(nS)$.
TL;DR: In this paper, the differential equations describing radiative transfer in vegetative canopies are generalized and solved to derive a layer scattering matrices, which can be applied to the calculation of optical parameters for multilayer ensembles according to the Adding method.
TL;DR: In this article, a uniform normal approximation for the distribution of the estimator of β is given, under which arbitrary linear combinations of β are asymptotically normal (when appropriately normalized).
Abstract: In a general linear model, $Y = X\beta + R$ with $Y$ and $R n$-dimensional, $X$ a $n \times p$ matrix, and $\beta p$-dimensional, let $\hat\beta$ be an $M$ estimator of $\beta$ satisfying $0 = \sum x_i\psi(y_i - x'_i\beta)$. Let $p \rightarrow \infty$ such that $(p \log n)^{3/2} /n \rightarrow 0$. Then $\max_i|x'_i(\hat{\beta} - \beta)| \rightarrow _P 0$, and it is possible to find a uniform normal approximation for the distribution of $\hat{\beta}$ under which arbitrary linear combinations $a'_n (\hat{\beta} - \beta)$ are asymptotically normal (when appropriately normalized) and $(\hat{\beta} - \beta)'(X'X)(\hat{\beta} - \beta)$ is approximately $\chi^2_p$.
TL;DR: A review of the literature on fiber-reinforced metal-matrix composites can be found in this paper, where fabrication methods, mechanical properties, secondary working techniques and interfaces are discussed.
TL;DR: In this article, the problem of anisotropy in substrate materials for microwave integrated-circuit applications is addressed, and several quasi-static, dynamic and empirical methods employed to obtain the propagation characteristics of microstrip, coplanar waveguides, and slotlines on anisotropic substrates are presented.
Abstract: This paper addresses the problem of anisotropy in substrate materials for microwave integrated-circuit applications. It is shown that in modeling the circuit characteristics, a serious error is incurred which becomes larger with increasing frequency when the substrate anisotropy is neglected. Quasi-static, dynamic, and empirical methods employed to obtain the propagation characteristics of microstrip, coplanar waveguides, and slotlines on anisotropic substrates are presented. Numerical solutions such as the method of moments and the transmission-line matrix technique are outlined. The modified Wiener-Hopf, the Fourier series techniques, and the method of lines are also discussed. A critique of the aforementioned methods and suggestions for future research directions are presented. The paper includes new results as well as a review of established methods.
TL;DR: In this article, the general state-space model for a 2D linear digital system is presented and a new definition of state-transition matrix is given based on the definition.
Abstract: The general state-space model for a 2-D linear digital system is presented. A new definition of state-transition matrix is given. Based on the definition, it is easy to calculate the state-transition matrix for any linear digital system. The general response formula for a system follows simply from the definition. A new definition of the characteristic function of a system and a theorem parallel to the Cayley-Hamilton theorem are also given. The presented results apply to any linear causal system.
Book•
01 Jan 1985
TL;DR: In this paper, Langer proposed the WKB method to solve the problem of total reflection and turning points without turning points, which is the basis for the present paper, and proved its correctness.
Abstract: I Historical Introduction.- 1.1. Early Asymptotic Theory Without Turning Points.- 1.2. Total Reflection and Turning Points.- 1.3. Hydrodynamic Stability and Turning Points.- 1.4. The So-Called WKB Method.- 1.5. The Contribution of R. E. Langer.- 1.6. Remarks on Recent Trends.- II Formal Solutions.- 2.1. Introduction.- 2.2. The Jordan Form of Holomorphic Functions.- 2.3. A Formal Block Diagonalization.- 2.4. Parameter Shearing: Its Nature and Purpose.- 2.5. Simplification by a Theorem of Arnold.- 2.6. Parameter Shearing: Its Application.- 2.7. Parameter Shearing: The Exceptional Case.- 2.8. Formal Solution of the Differential Equation.- 2.9. Some Comments and Warnings.- III Solutions Away From Turning Points.- 3.1. Asymptotic Power Series: Definition of Turning Points.- 3.2. A Method for Proving the Analytic Validity of Formal.- Solutions: Preliminaries.- 3.3. A General Theorem on the Analytic Validity of Formal.- Solutions.- 3.4. A Local Asymptotic Validity Theorem.- 3.5. Remarks on Points That Are Not Asymptotically Simple.- IV Asymptotic Transformations of Differential Equations.- 4.1. Asymptotic Equivalence.- 4.2. Formal Invariants.- 4.3. Formal Circuit Relations with Respect to the Parameter.- V Uniform Transformations at Turning Points: Formal Theory.- 5.1. Preparatory Simplifications.- 5.2. A Method for Formal Simplification in Neighborhoods of a Turning Point.- 5.3. The Case h > 1.- 5.4. The General Theory for n = 2.- VI Uniform Transformations at Turning Points: Analytic Theory.- 6.1. Preliminary General Results.- 6.2. Differential Equations Reducible to Airy's Equation.- 6.3. Differential Equations Reducible to Weber's Equation.- 6.4. Uniform Transformations in a Full Neighborhood of.- a Turning Point.- 6.5. Complete Reduction to Airy's Equation.- 6.6. Reduction to Weber's Equation in Wider Sectors.- 6.7. Reduction to Weber's Equation in a Full Disk.- VII Extensions of the Regions of Validity of the Asymptotic Solutions.- 7.1. Introduction.- 7.2. Regions of Asymptotic Validity Bounded by Separation Curves: The Problem.- 7.3. Solutions Asymptotically Known in Sectors Bounded by.- Separation Curves.- 7.4. Singularities of Formal Solutions at a Turning Point.- 7.5. Asymptotic Expansions in Growing Domains.- 7.6. Asymptotic Solutions in Expanding Regions: A General Theorem.- 7.7. Asymptotic Solutions in Expanding Regions: A Local Theorem.- VIII Connection Problems.- 8.1. Introduction.- 8.2. Stretching and Parameter Shearing.- 8.3. Calculation of the Restraint Index.- 8.4. Inner and Outer Solutions for a Particular nth-Order System.- 8.5. Calculation of a Central Connection Matrix.- 8.6. Connection Formulas Calculated Through Uniform Simplification.- IX Fedoryuk's Global Theory of Second-Order Equations.- 9.1. Global Formal Solutions of ?2u"=a(x)u2u" = a(x)u.- 9.2. Separation Curves for ?2u"=a(x)u2u" = a(x)u.- 9.3. A Global Asymptotic Existence Theorem for ?2u"=a(x)u2u" = a(x)u.- X Doubly Asymptotic Expansions.- 10.1. Introduction.- 10.2. Formal Solutions for Large Values of the.- Independent Variable.- 10.3. Asymptotic Solutions for Large Values of the.- Independent Variable.- 10.4. Some Properties of Doubly Asymptotic Solutions.- 10.5. Central Connection Problems in Unbounded Regions.- XI A Singularly Perturbed Turning Point Problem.- 11.1. The Problem.- 11.2. A Simple Example.- 11.3. The General Case: Formal Part.- 11.4. The General Case: Analytic Part.- XII Appendix: Some Linear Algebra for Holomorphic Matrices.- 12.1. Vectors and Matrices of Holomorphic Functions.- 12.2. Reduction to Jordan Form.- 12.3. General Holomorphic Block Diagonalization.- 12.4. Holomorphic Transformation of Matrices into Arnold's Form.- References.
TL;DR: In this article, a special class of composite local operators of any twist whose matrix elements between hadronic states can be expressed in terms of the free parton matrix elements averaged with hadron correlation functions was constructed.
TL;DR: In this article, a symmetric finite element method for solving fluid-structure interaction problems is presented, which uses velocity potentials and a hydrostatic pressure as unknowns in each fluid region, and displacements as knowns in the solid, and a C matrix (multiplied by time derivatives of nodal variables, but not a damping matrix) enforces the coupling between the variables.
TL;DR: Preliminary results indicate that higher quality or lower bit rates may be achieved with enough computational resources, and an extension of the centroid computation used in vector quantization is presented.
Abstract: Rate-distortion theory provides the motivation for using data compression techniques on matrices of N LPC vectors. This leads to a simple extension of speech coding techniques using vector quantization. The effects of using the generalized Lloyd algorithm on such matrices using a summed Itakura-Saito distortion measure are studied, and an extension of the centroid computation used in vector quantization is presented. The matrix quantizers so obtained offer substantial reductions in bit rates relative to full-search vector quantizers. Bit rates as low as 150 bits/s for the LPC matrix information (inclusive of gain, but without pitch and voicing) have been achieved for a single speaker, having average test sequence and codebook distortions comparable to those in the equivalent full-search vector quantizer operating at 350 bits/s. Preliminary results indicate that higher quality or lower bit rates may be achieved with enough computational resources.
TL;DR: In this paper, upper bound results for the plastic limit-loads are used to illustrate, on a quantitative basis, the truncation of the Berg-Gurson dilational-plastic yield locus, which was illustrated only on a qualitative basis.
TL;DR: In this paper, the authors propose a procedure for calculating the greatest lower bound for the asymptotic covariance matrices of the estimators of the generalized method of moments estimators.
01 Dec 1985
TL;DR: This paper describes parallel algorithms that have good numerical stability and remain efficient as n grows large and describes a quadratically convergent iterative method that gives the inverse of an n × n rational matrix A, the first known polylog time algorithm that is numerically stable.
Abstract: The most efficient known parallel algorithms for inversion of a nonsingular n × n matrix A or solving a linear system Ax = b over the rationals require O(log n)2 time and M(n)n0.5 processors (where M(n) is the number of processors required in order to multiply two n × n rational matrices in time O(log n).) Furthermore, all known polylog time algorithms for those problems are unstable: they require the calculation to be done with perfect precision; otherwise they give no results at all.This paper describes parallel algorithms that have good numerical stability and remain efficient as n grows large. In particular, we describe a quadratically convergent iterative method that gives the inverse (within the relative precision 2-nO(1)) of an n × n rational matrix A with condition ≤ n0(1) in O(log n)2 time using M(n) processors. This is the optimum processor bound and the factor n0.5 improvement of known processor bounds for polylog time matrix inversion. It is the first known polylog time algorithm that is numerically stable. The algorithm relies on our method of computing an approximate inverse of A that involves O(log n) parallel steps and n2 processors.Also, we give a parallel algorithm for solution of a linear system Ax = b with a sparse n × n symmetric positive definite matrix A. If the graph G(A) (which has n vertices and has an edge for each nonzero entry of A) is s(n)-separable, then our algorithm requires only O((log n)(log s(n))2) time and |E| + M(s(n)) processors. The algorithm computes a recursive factorization of A so that the solution of any other linear system Ax = b′ with the same matrix A requires only O(log n log s(n)) time and |E| + s(n)2 processors.
TL;DR: In this paper, an analytical procedure for the static shape control of flexible space structures subjected to thermal distortions is developed which is based on prescribing temperatures in control elements having much higher coefficients of thermal expansion than the main structure.
Abstract: An analytical procedure for the static shape control of flexible space structures subjected to thermal distortions is developed which is based on prescribing temperatures in control elements having much higher coefficients of thermal expansion than the main structure. The temperatures at the control elements are defined so as to minimize the overall thermal distortion of the structure from its ideal shape, and a matrix equation is obtained which can be solved for the set of optimum control temperatures. A formulation of the procedure for continuous structures governed by differential equations and a formulation for discrete (finite element modeled) structures governed by matrix equations are presented. The equations from the continuous formulation are employed for the shape control of a simple beam distorted by nonuniform heating, and the discrete formulation is applied in a general purpose finite-element structural analysis computer program for the shape control of a 750 m radiometer antenna reflector dish subjected to orbital heating. A reduction in thermal distortion by a factor of nearly 50 was obtained with the use of only seven control elements. Results for four different sets of control locations for the antenna are presented in which reductions in distortion of up to a factor of four were obtained.
TL;DR: A polynomial-time algorithm is presented to transform a totally-balanced matrix into a greedy matrix as well as to recognize a totallybalanced matrix, which enables us to solve a class of integer programming problems defined on totally- balanced matrices.
Abstract: Totally-balanced and greedy matrices are $( 0,1 )$-matrices defined by excluding certain submatrices. For a $n \times m \,( 0,1)$-matrix A we show that the linear programming problem $\max \{ by \mid yA\leqq c,0\leqq y\leqq d \}$ can be solved by a greedy algorithm for all $c\geqq 0$, $d\geqq 0$ and $b_1 \geqq b_2 \geqq \cdots \geqq b_n \geqq 0$, if and only if A is a greedy matrix. Furthermore we show constructively that if b is an integer, then the corresponding primal problem $\min \{ cx + dz \mid Ax + z\geqq b,x\geqq 0,z\geqq 0 \}$ has an integer optimal solution. A polynomial-time algorithm is presented to transform a totally-balanced matrix into a greedy matrix as well as to recognize a totallybalanced matrix. This transformation algorithm together with the result on greedy matrices enables us to solve a class of integer programming problems defined on totally-balanced matrices. Two examples arising in tree location theory are presented.
TL;DR: In this article, the adjoint problem of estimating the covariance of predicted heads in a finite element model with uncertain parameters is studied in a two-dimensional linear triangular finite element (TFE) model.
Abstract: Finite difference and finite element methods are frequently used to study aquifer flow; however, additional analysis is required when model parameters, and hence predicted heads are uncertain. Computational algorithms are presented for steady and transient models in which aquifer storage coefficients, transmissivities, distributed inputs, and boundary values may all be simultaneously uncertain. Innovative aspects of these algorithms include a new form of generalized boundary condition; a concise discrete derivation of the adjoint problem for transient models with variable time steps; an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation; and a new efficient first-order second-moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values. The techniques are presented in matrix form, and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations. Details of matrix structures are provided for a two-dimensional linear triangular finite element model.
TL;DR: Algorithms whose resource needs grow only linearly with problem size are developed, which will fully exploit the ‘parallel-processing’ capability available in the new generation of multi-processor computers.
TL;DR: In this article, a generalization of the matrix Bethe ansatz to systems with complicated pseudovacuum is proposed and the eigenvalues of the corresponding transfer matrices on a finite lattice are calculated.
Abstract: The authors investigate quantum models on a chain with O(n) and Sp(2k) symmetry. The eigenvalues of the corresponding transfer matrices on a finite lattice are calculated. A generalization of the matrix Bethe ansatz to systems with complicated pseudovacuum is proposed.