Showing papers on "Matrix (mathematics) published in 1990"

A General Mass-Conservative Numerical Solution for the Unsaturated Flow Equation

Abstract: Numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems. Numerical solution based on the standard h-based form of Richards equation generally yields poor results, characterized by large mass balance errors and erroneous estimates of infiltration depth. Conversely, numerical solutions based on the mixed form of Richards equation can be shown to possess the conservative property, so that mass is perfectly conserved. This leads to significant improvement in numerical solution performance, while requiring no additional computational effort. However, use of the mass-conservative method does not guarantee good solutions. Accurate solution of the unsaturated flow equation also requires use of a diagonal time (or mass) matrix. Only when diagonal time matrices are used can the solution be shown to obey a maximum principle, which guarantees smooth, nonoscillatory infiltration profiles. This highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow.

Coupled cluster response functions

Abstract: The linear and quadratic response functions have been determined for a coupled cluster reference state From the response functions, computationally tractable expressions have been derived for excitation energies, first‐ and second‐order matrix transition elements, transition matrix elements between excited states, and second‐ and third‐order frequency‐dependent molecular properties

SPARSKIT: A basic tool kit for sparse matrix computations

Abstract: Presented here are the main features of a tool package for manipulating and working with sparse matrices. One of the goals of the package is to provide basic tools to facilitate the exchange of software and data between researchers in sparse matrix computations. The starting point is the Harwell/Boeing collection of matrices for which the authors provide a number of tools. Among other things, the package provides programs for converting data structures, printing simple statistics on a matrix, plotting a matrix profile, and performing linear algebra operations with sparse matrices.

Truncated Singular Value Decomposition Solutions to Discrete Ill-Posed Problems with Ill-Determined Numerical Rank.

Abstract: Tikhonov regularization is a standard method for obtaining smooth solutions to discrete ill-posed problems. A more recent method, based on the singular value decomposition (SVD), is the truncated SVD method. The purpose of this paper is to show, under mild conditions, that the success of both truncated SVD and Tikhonov regularization depends on satisfaction of a discrete Picard condition, involving both the matrix and the right-hand side. When this condition is satisfied, then both methods are guaranteed to produce smooth solutions that are very similar.

Lower bounds for parametric estimation with constraints

Abstract: A Chapman-Robbins form of the Barankin bound is used to derive a multiparameter Cramer-Rao (CR) type lower bound on estimator error covariance when the parameter theta in R/sup n/ is constrained to lie in a subset of the parameter space. A simple form for the constrained CR bound is obtained when the constraint set Theta /sub C/, can be expressed as a smooth functional inequality constraint. It is shown that the constrained CR bound is identical to the unconstrained CR bound at the regular points of Theta /sub C/, i.e. where no equality constraints are active. On the other hand, at those points theta in Theta /sub C/ where pure equality constraints are active the full-rank Fisher information matrix in the unconstrained CR bound must be replaced by a rank-reduced Fisher information matrix obtained as a projection of the full-rank Fisher matrix onto the tangent hyperplane of the full-rank Fisher matrix onto the tangent hyperplane of the constraint set at theta . A necessary and sufficient condition involving the forms of the constraint and the likelihood function is given for the bound to be achievable, and examples for which the bound is achieved are presented. In addition to providing a useful generalization of the CR bound, the results permit analysis of the gain in achievable MSE performance due to the imposition of particular constraints on the parameter space without the need for a global reparameterization. >

New decomposition of the radar target scattering matrix

Abstract: A new decomposition of the complex radar target scattering matrix resolves a 2 × 2 complex scattering matrix into three components that provide a clearer picture of the physical mechanisms behind the scattering and, thereby, a clearer picture of the target itself.

Critical exponents for the one-dimensional Hubbard model

Abstract: Using results on the scaling of energies with the size of the system and the principles of conformal quantum field theory, we calculate the asymptotics of correlation functions for the one-dimensional Hubbard model in the repulsive regime in the presence of an external magnetic field. The critical exponents are given in terms of a dressed charge matrix that is defined in terms of a set of integral equations obtained from the Bethe-Ansatz solution for the Hubbard model. An interpretation of this matrix in terms of thermodynamical coefficients is given, and several limiting cases are considered.

Matrices ' Methods and Applications'

Abstract: How matrices arise Basic algebra of matrices Unique solution of linear equations Determinant and inverse Rank, non-unique solution of equations, and applications Eigenvalues and eigenvectors Quadratic and hermitian forms Canonical forms Matrix functions Generalized inverses Polynomials, stability, and matrix equations Polynomial and rational matrices Patterned matrices Miscellaneous topics Bibliography Index

Pressure Transient Analysis of Fractal Reservoirs

Abstract: The authors present a formulation for a fractal fracture network embedded into a Euclidean matrix. Single-phase flow in the fractal object is described by an appropriate modification of the diffusivity equation. The system's pressure-transient response is then analyzed in the absence of matrix participation and when both the fracture network and the matrix participate. The results obtained extend previous pressure-transient and well-testing methods to reservoirs of arbitrary (fractal) dimensions and provide a unified description for both single- and dual-porosity systems. Results may be used to identify and model naturally fractured reservoirs with multiple scales and fractal properties.

Algebraic multilevel preconditioning methods, II

Abstract: A recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed. It is based on a sequence of nested finite element spaces with the usual nodal basis functions. Using a nodeordering corresponding to the nested meshes, the finite element stiffness matrix is recursively split up into two-level block structures and is factored approximately in such a way that any successive Schur complement is replaced (approximated) by a matrix defined recursively and thereform only implicitely given. To solve a system with this matrix we need to perform a fixed number (v) of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level. It is shown that by a proper choice of iteration parameters it suffices to use\(v > \left( {1 - \gamma ^2 } \right)^{ - \tfrac{1}{2}} \) iterations for the so constructedv-foldV-cycle (wherev=2 corresponds to aW-cycle) preconditioning matrices to be spectrally equivalent to the stiffness matrix. The conditions involve only the constant λ in the strengthened C.-B.-S. inequality for the corresponding two-level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance. If we use successive uniform refinements of the meshes the method is of optimal order of computational complexity, if\(\gamma ^2< \tfrac{8}{9}\). Under reasonable assumptions of the finite element mesh, the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method, for instance.

Matrix methods applied to acoustic waves in multilayers

Abstract: Matrix methods for analyzing the electroacoustic characteristics of anisotropic piezoelectric multilayers are described. The conceptual usefulness of the methods is demonstrated in a tutorial fashion by examples showing how formal statements of propagation, transduction, and boundary-value problems in complicated acoustic layered geometries such as those which occur in surface acoustic wave (SAW) devices, in multicomponent laminates, and in bulk-wave composite transducers are simplified. The formulation given reduces the electroacoustic equations to a set of first-order matrix differential equations, one for each layer, in the variables that must be continuous across interfaces. The solution to these equations is a transfer matrix that maps the variables from one layer face to the other. Interface boundary conditions for a planar multilayer are automatically satisfied by multiplying the individual transfer matrices in the appropriate order, thus reducing the problem to just having to impose boundary conditions appropriate to the remaining two surfaces. The computational advantages of the matrix method result from the fact that the problem rank is independent of the number of layers, and from the availability of personal computer software that makes interactive numerical experimentation with complex layered structures practical. >

A taxonomy for conjugate gradient methods

Abstract: The conjugate gradient method of Hestenes and Stiefel is an effective method for solving large, sparse Hermitian positive definite (hpd) systems of linear equations, $Ax = b$. Generalizations to non-hpd matrices have long been sought. The recent theory of Faber and Manteuffel gives necessary and sufficient conditions for the existence of a CG method. This paper uses these conditions to develop and organize such methods. It is shown that any CG method for $Ax = b$ is characterized by an hpd inner product matrix B and a left preconditioning matrix C. At each step the method minimizes the B-norm of the error over a Krylov subspace. This characterization is then used to classify known and new methods. Finally, it is shown how eigenvalue estimates may be obtained from the iteration parameters, generalizing the well-known connection between CG and Lanczos. Such estimates allow implementation of a stopping criterion based more nearly on the true error.

A comparative study of algorithms for matrix balancing

Abstract: The problem of adjusting the entries of a large matrix to satisfy prior consistency requirements occurs in economics, urban planning, statistics, demography, and stochastic modeling; these problems are called Matrix Balancing Problems. We describe five applications of matrix balancing and compare the algorithmic and computational performance of balancing procedures that represent the two primary approaches for matrix balancing-matrix scaling and nonlinear optimization. The algorithms we study are the RAS algorithm, a diagonal similarity scaling algorithm, and a truncated Newton algorithm for network optimization. We present results from computational experiments with large-scale problems based on producing consistent estimates of Social Accounting Matrices for developing countries.

Efficient algorithm for optimal force distribution-the compact-dual LP method

Abstract: An efficient algorithm, the compact-dual linear programming (LP) method, is presented to solve the force distribution problem. In this method, the general solution of the linear equality constraints is obtained by transforming the underspecified matrix into row-reduced echelon form; then, the linear equality constraints of the force distribution problem are eliminated. In addition, the duality theory of linear programming is applied. The resulting method is applicable to a wide range of systems, constraints, and objective functions and yet is computationally efficient. The significance of this method is demonstrated by solving the force distribution problem of a grasping system under development at Ohio State called DIGITS. With two fingers grasping an object and hard point contact with friction considered, the CPU time on a VAX-11/785 computer is only 1.47 ms. If four fingers are considered and a linear programming package in the IMSL library is utilized, the CPU time is then less than 45 ms. >

Consensus patterns in DNA.

Abstract: Matrices can provide realistic representations of protein/DNA specificity. In many cases simple mononucleotide-based matrices are adequate representations, but more complex matrices may be needed for other cases. Unlike simple consensus sequences, matrices allow for different penalties to be assessed for different changes to a binding site, a property that is essential for accurate description of a binding site pattern. When only a collection of binding site sequences is known, the best representation for the pattern is an information content formulation, based on both thermodynamic and statistical considerations. Quantitative data on relative binding affinities may be used to determine matrices that provide a best fit to the data. Matrix representations also provide an efficient method of aligning multiple sequences to identify binding site patterns that they have in common.

On the numerical evaluation of Eshelby's tensor and its application to elastoplastic fibrous composites

Abstract: A numerical evaluation of Eshelby's S tensor for an ellipsoidal inclusion imbedded in a general anisotropic matrix material is performed. The numerical scheme is valid for any degree of matrix anisotropy and for any aspect ratio of the ellipsoid, including the extreme cases of cracks and cylindrical inclusions. The influence of matrix anisotropy on the evaluation of S is tested extensively for cylindrical inclusions by considering plasticity induced anisotropy in the instantaneous properties of an elastic-plastic matrix material. The Mori-Tanaka averaging method is used to study the influence of the evaluation of S on the prediction of instantaneous effective properties of fibrous composites with elastic fibres and elastic-plastic matrix.

The overall elastoplastic stress-strain relations of dual-phase metals

Abstract: T wo simple , albeit approximate, theories are developed to estimate the stress-strain relations of dual-phase metals of the inclusion-matrix type, where both phases are capable of undergoing plastic flow. The first one is based upon Hill's recognition of a weakening constraint power in a plastically deforming matrix, whereas the second one is based on Kroner's elastic constraint in the treatment of the single inclusion-matrix interaction. The inclusion-inclusion interaction at finite concentration is accounted for by the Mori-Tanaka method in both cases. Consistent with the known elastic behavior, the first theory discloses that the geometrical arrangement of the constituents has a significant influence on the overall elastoplastic response. When the harder phase takes the position of the matrix the composite is far Stiffer than that when it takes the position of inclusions. The strong elastic constraint associated with the second theory tends to provide an upper-bound type of estimate regardless of whether the matrix is the harder phase or the softer, and, therefore, it is suggested that this theory be used only for the class of composites whose matrix is the harder phase. Both theories are finally applied to predict the stress-strain relations of dual-phase stainless steels, and the results are found to be in satisfactory agreement with the test data.

On homogeneous transforms, quaternions, and computational efficiency

Abstract: Three-dimensional modeling of rotations and translations in robot kinematics is most commonly performed using homogeneous transforms. An alternate approach, using quaternion-vector pairs as spatial operators, is compared with homogeneous transforms in terms of computational efficiency and storage economy. The conclusion drawn is that quaternion-vector pairs are as efficient as, more compact than, and more elegant than their matrix counterparts. A robust algorithm for converting rotational matrices into equivalent unit quaternions is described, and an efficient quaternion-based inverse kinematics solution for the Puma 560 robot arm is presented. >

Tunneling matrix elements in three-dimensional space: The derivative rule and the sum rule.

Abstract: In this paper, a systematic derivation of the tunneling matrix elements in three-dimensional space is presented. Based on a modified Bardeen tunneling theory, explicit expressions for the tunneling matrix elements for localized tip states are derived with use of the Green's-function method. It is shown that by expanding the vacuum tail of the tip wave function in terms of spherical harmonics, the tunneling matrix elements are related to the derivatives of the sample wave functions at the nucleus of the apex atom (taken as the center of the spherical-harmonics expansion), in a simple and straightforward way. In addition, an independent derivation based on a general sum rule is also presented, which is valid in a number of curvilinear coordinate systems. In spherical coordinates, a general form of the derivative rule follows. In parabolic coordinates, similar results are obtained. Physical meanings of these matrix elements, as well as their implications to the imaging mechanism of scanning-tunneling microscopy, are discussed.

Analysis of the Cholesky Decomposition of a Semi-definite Matrix

Abstract: Perturbation theory is developed for the Cholesky decomposition of an $n \times n$ symmetric positive semidefinite matrix $A$ of rank~$r$. The matrix $W=\All^{-1}\A{12}$ is found to play a key role in the perturbation bounds, where $\All$ and $\A{12}$ are $r \times r$ and $r \times (n-r)$ submatrices of $A$ respectively. A backward error analysis is given; it shows that the computed Cholesky factors are the exact ones of a matrix whose distance from $A$ is bounded by $4r(r+1)\bigl( orm{W}+1\bigr)^2u orm{A}+O(u^2)$, where $u$ is the unit roundoff. For the complete pivoting strategy it is shown that $ orm{W}^2 \le {1 \over 3}(n-r)(4^r- 1)$, and empirical evidence that $ orm{W}$ is usually small is presented. The overall conclusion is that the Cholesky algorithm with complete pivoting is stable for semi-definite matrices. Similar perturbation results are derived for the QR decomposition with column pivoting and for the LU decomposition with complete pivoting. The results give new insight into the reliability of these decompositions in rank estimation.