Showing papers on "Basis (linear algebra) published in 1985"
TL;DR: It is shown that patterns of impairments observed in brain-damaged patients support the major assumptions of the model and that the model provides a theoretically motivated framework for interpreting the deficits.
611 citations
TL;DR: In this article, a Rayleigh-Ritz analysis of the nonlinear system of ordinary differential equations resulting from the finite element discretization is reduced by means of Rayleigh Ritz analysis, and the basis vectors are chosen to be the current tangent eigenmodes together with some modal derivatives that indicate the way in which the spectrum is changing.
178 citations
TL;DR: The reduced basis error is shown to be dominated by an approximation error, which leads to error estimates for projection onto specific subspaced; for example, subspaces related to Taylor, Lagrange and discrete least-squares approximation.
Abstract: The reduced basis method is a projection technique for approximating the solution curve of a finite system of nonlinear algebraic equations by the solution curve of a related system that is typically of much lower dimension. In this paper, the reduced basis error is shown to be dominated by an approximation error. This, in turn, leads to error estimates for projection onto specific subspaces; for example, subspaces related to Taylor, Lagrange and discrete least-squares approximation.
161 citations
PARC1
TL;DR: In this paper, it was shown that any bounded region in Ed can be divided into 2d subregions of equal volume in such a way that no hyperplane in Ed cannot intersect all 2d of the sub-regions.
Abstract: It is shown that any bounded region in Ed can be divided into 2d subregions of equal volume in such a way that no hyperplane in Ed can intersect all 2d of the subregions. This theorem provides the basis of a data structure scheme for organizing n points in d dimensions. Under this scheme, a broad class of geometric queries in d dimensions, including many common problems in range search and optimization, can be solved in linear storage space and sublinear time.
142 citations
TL;DR: In contrast to "ordinary" sensitivity analysis in linear programming, the tolerance approach in this paper considers simultaneous and independent changes in the objective function coefficients and in the right-hand side terms and yields a maximum tolerance percentage such that, as long as selected coefficients or terms are accurate to within that percentage of their estimated values, the same basis is optimal.
Abstract: In constrast to "ordinary" sensitivity analysis in linear programming, the tolerance approach considers simultaneous and independent changes in the objective function coefficients and in the right-hand side terms. This approach yields a maximum tolerance percentage such that, as long as selected coefficients or terms are accurate to within that percentage of their estimated values, the same basis is optimal. In particular, if the objective function coefficients are accurate to within the maximum tolerance percentage of their specified values, then the same solution is optimal.
109 citations
TL;DR: In this article, a Rayleigh-Ritz technique is used to reduce the nonlinear system of ordinary differential equations obtained from the finite element discretization by employing the Rayleigh Ritz technique, and a new criterion for the computation of the basis vectors is proposed.
97 citations
TL;DR: In this paper, the exact Gaussian likelihood for structural parameters in nonstationary higher-order continuous-time dynamic models was derived and applied in the estimation of these parameters, which completely avoids the computation of the covariance matrix of the observations and is applicable to a system of any order with mixed stock and flow data.
Abstract: This paper is concerned with derivation of a new efficient algorithm for computing the exact Gaussian likelihood for structural parameters in nonstationary higher-order continuous-time dynamic models and with its application in the estimation of these parameters. The algorithm completely avoids the computation of the covariance matrix of the observations and is applicable to a system of any order with mixed stock and flow data. It is used as the basis for an iterative procedure in which the structural parameters and the initial state vector are estimated alternately.
89 citations
TL;DR: In this article, the time-domain integral equation for the three-dimensional vector electric field is formulated as a convolution of the scattering current with the tensor Green's function.
Abstract: The time‐domain integral equation for the three‐dimensional vector electric field is formulated as a convolution of the scattering current with the tensor Green’s function. The convolution integral is divided into a sum of integrals over successive time steps, so that a numerical scheme can be formulated with a time stepping approximation of the convolution of past values of the solution with the system impulse response. This, together with spatial discretization, leads to a matrix equation in which previous solution vectors are multiplied by a series of matrices and fed back into the system by adding to the primary field source vector. The spatial discretization, based on a modification of the usual pulse basis formulation in the frequency domain, includes an additional subset of divergence‐free basis functions generated by integrating the Green’s function around concentric closed rectangular paths. The inductive response of the body is more accurately modeled with these additional basis functions, and a...
73 citations
TL;DR: A general algorithm for low-order multifunctional observer design with arbitrary eigenvalues which can generate a functional observer with different orders which are no larger but usually much less than m(v - 1), wheremis the number of functionals andvis the observability index of(A, C).
Abstract: This paper presents a general algorithm for low-order multifunctional observer design with arbitrary eigenvalues. The feature of this algorithm is that it can generate a functional observer with different orders which are no larger but usually much less than m(v - 1) , where m is the number of functionals and v is the observability index of (A, C) . Since the order needed for the observer varies with the functionals besides other system parameters, this design approach should be practical. The resulting observer system matrix is in its Jordan form. The key step of this algorithm is the generation of the basis for the transformation matrix which relates the system and observer states. The computation of this algorithm is quite reliable. It is based on the block observable lower Hessenberg form of (A, C) , and all its initial and major computation involves only the orthogonal operations.
70 citations
TL;DR: This paper describes and test a variation of a method originally suggested by Topcu and called the turnback algorithm for computing a banded basis matrixB, and results indicate that both implementations of the algorithm yielded a well-conditioned, banded, basis matrix B when A is well- Conditioned.
Abstract: LetA be a realm×n matrix with full row rankm. In many algorithms in engineering and science, such as the force method in structural analysis, the dual variable method for the Navier-Stokes equations or more generally null space methods in quadratic programming, it is necessary to compute a basis matrixB for the null space ofA. HereB isn×r, r=n?m, of rankr, withAB=0. In many instancesA is large and sparse and often banded. The purpose of this paper is to describe and test a variation of a method originally suggested by Topcu and called the turnback algorithm for computing a banded basis matrixB. Two implementations of the algorithm are given, one using Gaussian elimination and the other using orthogonal factorization by Givens rotations. The FORTRAN software was executed on an IBM 3081 computer with an FPS-164 attached array processor at the Triangle Universities Computing Center and on a CYBER 205 vector computer. Test results on a variety of structural analysis problems including two- and three-dimensional frames, plane stress, plate bending and mixed finite element problems are discussed. These results indicate that both implementations of the algorithm yielded a well-conditioned, banded, basis matrixB whenA is well-conditioned. However, the orthogonal implementation yielded a better conditionedB for large, illconditioned problems.
TL;DR: In this paper, the use of an orthogonal set of specially selected Ritz vectors is shown to be very effective in reducing the cost of dynamic analysis by modal superposition Several mechanical structures are examined, and the Ritz vector approach is compared to the classical eigenvector approach on the basis of cost, accuracy and elapsed analysis (throughput) time.
TL;DR: This paper describes a method for detecting persons walking in a passageway from two consecutive TV images at long intervals by extracting candidate areas for persons from TV images by image processing.
TL;DR: The robust servomechanism theory is applied to the linear system that results when the overall, nonlinear, dynamic system is split, in the standard manner, into a nominal system and a (linear) system linearized about the nominal as discussed by the authors.
Abstract: This paper describes a framework for synthesizing control laws for manipulators based on robust servomechanism theory for multivariable linear systems. This framework takes into account the coupled and nonlinear nature of the differen tial equations describing the manipulator as well as the fact that the inputs and outputs are subject to large excursions.The robust servomechanism theory is applied to the linear system that results when the overall, nonlinear, dynamic system is split, in the standard manner, into a nominal sys tem and a (linear) system linearized about the nominal. A control law for the linear system is then derived on the basis of linear quadratic regulator theory. To ensure good dynamic response, the implicit model-following technique is used to choose the weights in the resulting performance index.The theory is then applied to design a control law for a two-degree-of-freedom spatial manipulator following a pre scribed trajectory. The effect of changing the speed and iner tias of the man...
TL;DR: In this article, expansions for each fundamental basis of the hydrogen atom over two others are found and an additional integral of motion corresponding to an elliptic basis is determined. But these expansions do not cover the full hydrogen atom.
Abstract: Expansions for each fundamental basis of the hydrogen atom over two others are found and an additional integral of motion corresponding to an elliptic basis is determined. Rcpresentations of the elliptic basis as a superposition of polar and parabolic states are obtained. Certain interesting limiting cases are investigated.
TL;DR: In this paper, a relativistic many-body perturbation theory of molecular electronic structure is discussed, and the choice of basis sets and the handling of negative energy states are discussed.
Abstract: For pt.II see ibid., vol.17, no.7, p.1201 (1984). The solution of the Dirac equation for hydrogen-like atoms within the algebraic approximation, that is, by using finite basis sets, is considered. The model problem of a hydrogenic atom with nuclear charge Z perturbed by a potential-Z'/r is used to demonstrate the feasibility of including relativistic effects in diagrammatic perturbation theory calculations performed within the algebraic approximation. Both ground- and excited state calculations are reported. Particular attention is paid to the choice of basis sets and the handling of the negative-energy states. It is shown that the 'finite basis set disease' can be avoided in perturbation theory calculations by making an appropriate choice of basis sets. Negative-energy states must be included in the perturbation theory summations. The importance of this work to the development of a relativistic many-body perturbation theory of molecular electronic structure is discussed.
TL;DR: The paper raises a number of questions which could form the basis for useful research in this area and some suggestions for practical algorithms outlined.
TL;DR: The authors give necessary and sufficient conditions in order that the infinite product or sum of the terms of a positive decreasing sequence generates the reals in a given interval.
Abstract: We give necessary and sufficient conditions in order that the infinite product or sum of the terms of a positive decreasing sequence generates the reals in a given interval.
TL;DR: How Z can be obtained by updating an explicit QR factorization with Householder transformations is described and why the chosen form ofZ is convenient in certain methods for nonlinearly constrained optimization is indicated.
Abstract: Given a rectangular matrix A(x) that depends on the independent variables x, many constrained optimization methods involve computations with Z(x), a matrix whose columns form a basis for the null space of A/sup T/(x). When A is evaluated at a given point, it is well known that a suitable Z (satisfying A/sup T/Z = 0) can be obtained from standard matrix factorizations. However, Coleman and Sorensen have recently shown that standard orthogonal factorization methods may produce orthogonal bases that do not vary continuously with x; they also suggest several techniques for adapting these schemes so as to ensure continuity of Z in the neighborhood of a given point. This paper is an extension of an earlier note that defines the procedure for computing Z. Here, we first describe how Z can be obtained by updating an explicit QR factorization with Householder transformations. The properties of this representation of Z with respect to perturbations in A are discussed, including explicit bounds on the change in Z. We then introduce regularized Householder transformations, and show that their use implies continuity of the full matrix Q. The convergence of Z and Q under appropriate assumptions is then proved. Finally, we indicate why themore » chosen form of Z is convenient in certain methods for nonlinearly constrained optimization.« less
TL;DR: The extended set partitioning model forms the basis of a computer assisted bus crew scheduling system developed by the authors and is in regular use by Dublin City Services in the Republic of Ireland.
TL;DR: In this article, the authors examined the clustering of nations using different clustering methods and found divergent results in terms of reconciling these differences, the need for additional research comparing competitive and alternative interpretations of Smallest Space Analysis (SSA), and the use of independent and objective methods, e.g., cluster analysis.
Abstract: The present study examined the clustering of nations using different clustering methods. Organizational attitudes and perceptions of 1768 managers from IS Western nations employed by a multinational corporation were surveyed. Divergent results were found and are discussed in terms of reconciling these differences, the need for additional research comparing competitive and alternative interpretations of Smallest Space Analysis (SSA), and the use of independent and objective methods, e.g., cluster analysis.
TL;DR: In this paper, analytical derivative techniques were used to obtain the dipole moment derivatives and harmonic frequencies of C 2H 2 and C 2 H 4 using SCF wavefunctions and large basis sets.
TL;DR: In this paper, higher order multipole moments for the ground states of the HF, CO and N2 molecules are calculated using the matrix Hartree-Fock ansatz and distributed distributed multipole analyses are also reported.
Abstract: Higher order multipole moments for the ground states of the HF, CO and N2 molecules are calculated using the matrix Hartree-Fock ansatz. Calculations are reported for a universal systematic sequence of even-tempered basis sets of cartesian gaussian-type functions and the behaviour of the calculated higher order multipole moments with increasing size of basis set is examined. Distributed multipole analyses are also reported.
01 Jan 1985
TL;DR: This paper shows that the topology of the master basis tree and the rules by which it can admissibly be restructured can be characterized by seven mutually exclusive and collectively exhaustive basis exchange cases.
Abstract: The simplex special ordered network (SON) algorithm is a partitioning method for solving LP problems with embedded network structure. The algorithm derives from a theoretical characterization of the network topology of the basis embodied in a specially constructed master basis tree. In this paper we show that the topology of the master basis tree and the rules by which it can admissibly be restructured can be characterized by seven mutually exclusive and collectively exhaustive basis exchange cases. Further, these seven cases will always keep the network portion of the basis at its maximum dimension.
TL;DR: In this article, a canonical orthonormal basis for generic representations of the group chain SU(3) contains/implies SO(3), which is a generalization of the canonical basis for SO(2).
Abstract: A canonical orthonormal basis is given for generic representations of the group chain SU(3) contains/implies SO(3).
TL;DR: In this article, the effect of incompleteness on the deformation density of CO is studied by comparing various STO basis sets with a fully numerical (basis-free) result.
Abstract: The effect of basis set incompleteness on the deformation density of CO is studied by comparing various STO basis sets with a fully numerical (basis-free) result. A triple-zeta s, p basis plus one 3d and one 4f function appears to be practically converged. The convergence characteristics of other properties (Re, De, ωe, μ0, μ1, electric field gradient (EFG)) with respect to basis set size and type are also investigated. The convergence behaviour is similar for these properties and the deformation densities.
TL;DR: In this paper, an orthogonal series estimator is proposed, and Hilbert space methods are used in the derivation of its properties and the proof of several convergence theorems, which can be used for determining the number of Fourier coefficients to be estimated from a given sample.
Abstract: This paper is concerned with the estimation of a nonlinear regression function which is not assumed to belong to a prespecified parametric family of functions. An orthogonal series estimator is proposed, and Hilbert space methods are used in the derivation of its properties and the proof of several convergence theorems. One of the main objectives of the paper is to provide the theoretical basis for a practical stopping rule which can be used for determining the number of Fourier coefficients to be estimated from a given sample.
TL;DR: In this article, the authors extend their previous quantitative condensation principles to some nonlinear situations and show that the sharpness of error bounds for Fejer means on the quasinormed spaces L2Πq, 0 < q < l, 0
Abstract: In this paper we extend our previous quantitative condensation principles to some nonlinear situations. More specifically, on the basis of the usual homogeneity, we are interested in some reductions of the additivity which will in particular enable us to treat condensation on arbitrary point sets. The usefulness of the general result is illustrated by some first applications concerned with the sharpness of error bounds for Fejer means on the quasinormed spaces L2Πq, 0
TL;DR: In this article, a comparison of numerical methods with basis set expansion techniques in the evaluation of the sum-over-states expressions which arise in the diagrammatic perturbation expansion in second and higher orders is made.
Abstract: A comparison has been made of numerical methods with basis set expansion techniques in the evaluation of the sum-over-states expressions which arise in the diagrammatic perturbation expansion in second and higher orders. The model problem of a ground-state hydrogenic atom with charge Z perturbed by the potential -Z'/r is used. The energy corresponding to each diagrammatic component can be evaluated explicitly for this mode. Universal systematic sequences of even-tempered basis sets of exponential-type functions and Gaussian-type functions are employed and the convergence of the individual components of the energy expansion with increasing size of basis set is examined.
TL;DR: In this article, the Hamiltonian structure of nonlinear evolution equations for which the potentials of the Zakharov-Shabat system depend polynomially on a spectral parameter is investigated on the basis of a general group theoretic scheme.
Abstract: The Hamiltonian structure of nonlinear evolution equations for which the potentials of the Zakharov-Shabat system depend polynomially on a spectral parameter is investigated on the basis of a general group theoretic scheme. The orbits of the corresponding coadjoint action are calculated. Formulas for the generating functions of the densities and flows of conservation laws andM-operators are derived.