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Showing papers on "Basis (linear algebra) published in 1988"


Journal ArticleDOI
TL;DR: In this paper, the authors used double zeta plus polarization level atomic pair natural orbital basis sets to calculate molecular self-consistent field (SCF) energies and correlation energies.
Abstract: The major source of errror in most ab initio calculations of molecular energies is the truncation of the one‐electron basis set. A complete basis set model chemistry is defined to include corrections for basis set truncation errors. This model uses double zeta plus polarization level atomic pair natural orbital basis sets to calculate molecular self‐consistent‐field (SCF) energies and correlation energies. The small corrections to give the complete basis set SCF energies are then estimated using the l−6 asymptotic convergence of the multicenter angular momentum expansion. The calculated correlation energies of the atoms He, Be, and Ne, and of the hydrides LiH, BH3, CH4, NH3, H2O, and HF, using the double zeta plus polarization basis sets vary from 83.0% to 91.2% of the experimental correlation energies. However, extrapolation of each of the pair energies and pair‐coupling terms to the complete basis set values using the asymptotic convergence of pair natural orbital expansions retrieves from 99.5±0.7% to ...

2,329 citations


Journal ArticleDOI
TL;DR: Construction of B-spline basis sets for the Dirac-Hartree-Fock equations is described and the resulting basis sets are applied to study the cesium spectrum.
Abstract: A procedure is given for constructing basis sets for the radial Dirac equation from B splines. The resulting basis sets, which include negative-energy states in a natural way, permit the accurate evaluation of the multiple sums over intermediate states occurring in relativistic many-body calculations. Illustrations are given for the Coulomb-field Dirac equation and tests of the resulting basis sets are described. As an application, relativistic corrections to the second-order correlation energy in helium are calculated. Another application is given to determine the spectrum of thallium (where finite--nuclear-size effects are important) in a model potential. Construction of B-spline basis sets for the Dirac-Hartree-Fock equations is described and the resulting basis sets are applied to study the cesium spectrum.

421 citations


Journal ArticleDOI
01 May 1988
TL;DR: In a new method for automatic indexing and retrieval, implicit higher-order structure in the association of terms with documents is modeled to improve estimates of term-document association, and therefore the detection of relevant documents on the basis of terms found in queries.
Abstract: In a new method for automatic indexing and retrieval, implicit higher-order structure in the association of terms with documents is modeled to improve estimates of term-document association, and therefore the detection of relevant documents on the basis of terms found in queries. Singular-value decomposition is used to decompose a large term by document matrix into 50 to 150 orthogonal factors from which the original matrix can be approximated by linear combination; both documents and terms are represented as vectors in a 50- to 150- dimensional space. Queries are represented as pseudo-documents vectors formed from weighted combinations of terms, and documents are ordered by their similarity to the query. Initial tests find this automatic method very promising.

411 citations


Journal ArticleDOI
TL;DR: The hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements is derived and analyzed.
Abstract: We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauβ-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

318 citations


Journal ArticleDOI
TL;DR: In this article, the 16th degree polynomial input-output equation in the ten-half-angle of the output angular displacement for the general spatial 7-link 7R mechanism was derived.

177 citations


PatentDOI
Ira A. Gerson1
TL;DR: In this article, an improved vector generation and search technique was described for a code-excited linear prediction (CELP) speech coder using a codebook of excitation code vectors.
Abstract: An improved excitation vector generation and search technique (FIG. 1) is described for a code-excited linear prediction (CELP) speech coder (100) using a codebook of excitation code vectors. A set of M basis vectors Vm (n) are used along with the excitation signal codewords (i) to generate the codebook of excitation vectors ui (n) according to a "vector sum" technique (120) of converting the selector codewords into a plurality of interim data signals, multiplying the set of M basis vectors by the interim data signals, and summing the resultant vectors to produce the set of 2M codebook vectors. The entire codebook of 2M possible excitation vectors is efficiently searched by using the vector sum generation technique with the M basis vectors--without ever having to generate and evaluate each of the 2M code vectors themselves. Furthermore, only M basis vectors need to be stored in memory (114), as opposed to all 2M code vectors.

123 citations


Journal ArticleDOI
TL;DR: In this paper, a new vector theory for the analysis of spatial mechanisms was developed on the basis of vector analysis theory and dual-number algebra and the recursive notation presented by J. Duffy.

105 citations



Journal ArticleDOI
TL;DR: In this paper, the concept of preservation of harmonic analyticity is applied to find unconstrained prepotentials of hyper-Kahler geometry, and the results of the analysis are shown to be compatible with off-shell d = 4, N = 2 supersymmetric σ models.

67 citations


Journal ArticleDOI
TL;DR: In this article, a finite element based iterative method was introduced for the solution of the eigenvalue problem of stationary cracks, and the theoretical basis of this iterative approach was given.
Abstract: In a recently published paper a finite element based iterative method was introduced for the solution of the eigenvalue problem of stationary cracks.1 In this paper we give the theoretical basis of this iterative method and we show why it converges and how it could be extended to more complex fracture problems. The cases of cracks at interfaces are illustrated.

49 citations


Journal ArticleDOI
TL;DR: Enter conditions are established which guarantee the persistency of excitation of a large class of regression vectors obtained from both time-invariant and time-varying systems.
Abstract: For continuous-time, multiple-input, multiple-output, linear systems, we present conditions under which the persistency of excitation of one regression vector implies the persistency of another regression vector derived from the first via a linear, dynamical transformation. We then introduce a definition of sufficient richness for vector input signals in the form of a persistency of excitation condition on a basis regression vector. Finally we establish input conditions which guarantee the persistency of excitation of a large class of regression vectors obtained from both time-invariant and time-varying systems.

Journal ArticleDOI
TL;DR: The Husimi function is related to the Husimi matrix in a way analogous to the relationship between a density matrix and a density, but there is a major difference in that this map can always be inverted as discussed by the authors.
Abstract: The Husimi function provides a phase‐space view of quantum systems. This paper considers a number of its properties, including ways in which it can be expanded in terms of basis functions. A spectral or ‘‘natural’’ expansion and an expansion analogous to the Carlson–Keller expansion in terms of coordinate‐density momentum‐density products are considered, as is a method for separating the angular dependence of the momentum. There is a set of functions in phase space having the same overlap properties as the initial orbital basis in terms of which the charge density matrix is expressed. A Husimi matrix is defined and a scalar product in the space containing such matrices as elements is introduced. The connection with the vector space of density matrices is examined. The Husimi function is related to the Husimi matrix in a way analogous to the relationship between a density matrix and a density, but there is a major difference in that this map can always be inverted. For a harmonic oscillator basis the phase space basis functions, in terms of which the Husimi function is expressed, span a linear space but do not provide a complete set; their products provide a linearly independent set that is complete. It is suggested that similar behavior can be expected for other basis sets.

Journal ArticleDOI
TL;DR: In this article, a nonlinear transformation performs the Levy-constrained search formulation of the density functional for the electronic energy through a minimization of the energy with respect to a set of variational coefficients.
Abstract: A nonlinear transformation performs the Levy-constrained search formulation of the density functional for the electronic energy through a minimization of the energy with respect to a set of variational coefficients. The construction requires a complete set of arbitrary functions as the auxiliary basis. Truncation of the basis set provides an upper bound to the energy functional. Practical approaches to obtain accurate upper bounds to this functional are discussed, and a density-functional alternative to the standard Hartree-Fock method is described.

Proceedings ArticleDOI
07 Dec 1988
TL;DR: In this article, the problem of approximate linearization of nonlinear control systems is reduced to the solution of a set of linear equations, and a least-squares solution is proposed that minimizes in a statistical sense the error in the approximation.
Abstract: A method is presented to solve the approximate linearization problem of nonlinear control systems. The problem is reduced to the solution of a set of linear equations as follows. First, the generalized homological equations are derived. By introducing an appropriate basis for expressing higher degree monomials in the vector field, a set of equations linear in the coefficients of the monomials are found. An exact solution to this set of equations is not always possible. A least-squares solution is proposed that minimizes in a statistical sense the error in the approximation. >

Proceedings Article
01 Aug 1988

Journal ArticleDOI
15 May 1988
TL;DR: In this paper, the effects of basis set choice and level of inclusion of correlation on the prediction of the energy difference between the linear and triangular forms of SiC 2 are discussed.
Abstract: The effects of basis set choice and level of inclusion of correlation on the prediction of the energy difference between the linear and triangular forms of SiC 2 are discussed. It is concluded that at the SCF level the ordering is dependent on basis set choice and that in fact, contrary to earlier belief, a carefully optimized basis set may predict the triangular form to be slightly more stable even at the SCF level. However, correlation is essential to ensure the expected energy difference. Improvement of the basis set causes opposing changes in the absolute SCF and correlation energy difference between the two forms such that the total energy difference remains small. Estimating total energies by adding correlation effects calculated from small basis sets to results of largebasis SCF calculation is thus not valid. The energy difference between the linear and the triangular forms is 3.74 mh (2.35 kcal/mole) in MBPT (4) using the largest, 120 CGTO basis set. It is argued that inclusion of higher-order correlation may lead to further stabilization of the triangular form.

Journal ArticleDOI
TL;DR: The XFPS algorithm as mentioned in this paper uses cross vectors for finding relationships among the peaks of the symmetry minimum function, which suppress false peaks and reveal the correct solution with greater probability, in contrast to many other Patterson methods no a priori structural information is necessary.
Abstract: On the basis of a generalized symmetry minimum function several computer-oriented methods for interpreting Patterson functions and for locating the position of heavy-atom fragments in crystals belonging to space groups of higher symmetry than P1 have been developed. The methods utilize cross vectors for finding relationships among the peaks of the symmetry minimum function. This approach has the advantage of suppressing false peaks of the symmetry minimum function, in locating more than one atom and in revealing the correct solution with greater probability. The heavy-atom fragment can be extended by superposition or Fourier methods. The methods are valid for all space groups, are simple to apply and form the basis for fully automated structure determination. In contrast to many other Patterson methods no a priori structural information is necessary. A few selected examples demonstrate the power of the new version of the computer program XFPS.

Patent
10 Jun 1988
TL;DR: In this paper, a compensating beamformer is provided which comprises a plurality of sensing elements and an analog-to-digital converters for converting incoming analog signals to digital form.
Abstract: A compensating beamformer which requires orders of magnitude fewer calculations that prior art methods. A compensating beamformer is provided which comprises a plurality of sensing elements and a plurality of analog-to-digital converters, for converting incoming analog signals to digital form. Digital signals from at least four such elements are used to compute phase angle information which is combined to form a matrix of input data in the frequency domain. An unweighted steering vector is determined to sample data from the target direction. A corrector matrix is calculated based on input data from sensing elements. That corrector matrix along with its inverse, which is determined recursively, is used in combination with the unweighted steering vector to determine an optimal steering vector. The input data in the frequency domain are then multiplied by the optimal steering vector to obtain signals in the directions of interest. In preferred embodiments of this invention these calculations are repeated systolically to provide optimal steering vector updates on an essentially real-time basis.

Journal ArticleDOI
Pierre Léger1
TL;DR: In this paper, a load dependent transformation vector is proposed for dynamic response analysis of large structures by vector superposition, which is an economic alternative to the usual mode superposition method.

Journal ArticleDOI
TL;DR: In this paper, a general tree T is defined as a tree with arbitrary width and height any ordinal and it is shown that T has the metric approximation, Radon-Nikodym, and π properties.

Journal ArticleDOI
TL;DR: In this article, the authors presented a method of numerically solving the multichannel Schrodinger equation by propagating exact first-order coupled equations for a specially designed half collision matrix X(r).
Abstract: We present methods of numerically solving the multichannel Schrodinger equation by propagating exact first‐order coupled equations for a specially designed half collision matrix X(r). The method requires choosing a convenient set of reference potentials with which to generate a pair of reference radial functions for each channel. It is easy to tailor the choice of basis to the nature of the exact multichannel interaction matrix in a given region of space, either to enhance the numerical efficiency of propagating the exact X(r), or, to facilitate the introduction of useful approximations. In particular, we define a classical half collision matrix Z(r) by neglecting rapidly oscillating terms in the propagation of X(r). This random phase approximation can only be justified in classically accessible regions of r, and results in a set of first order equations which imposes the unitarity of Z(r)Z°(r)=1 throughout the propagation. Fortunately, such regions often contain the dominant couplings, particularly for h...

Journal ArticleDOI
TL;DR: In this paper, the density matrix was computed from the density using basis orbitals which form linearly independent products (LIP) bases containing several of the natural spin orbitals from an accurate nonrelativistic 650-term configuration interaction (CI) wavefunction.
Abstract: The density matrix was computed from the density using basis orbitals which form linearly independent products (LIP). Calculations were performed on the Be atom using LIP bases containing several of the natural spin orbitals from an accurate nonrelativistic 650-term configuration interaction (CI) wavefunction. Calculations were also performed using a basis set of 6 s-type orbitals optimized so that the lowest eigenvalue, d1, of the overlap matrix of products is maximized and the energy of a small CI wavefunction is minimized. The value of d1 in the optimized basis is 1 × 10−7. The density and the potential energy obtained from the 650-term CI wavefunction were accurately reproduced by the density matrix expressed in the optimized LIP basis, but the kinetic energy was somewhat less accurate.

Journal ArticleDOI
TL;DR: In this paper, a fully recursive method for identifying, estimat-ing and forecasting multivariate (vector) time-series is described, where any low frequency (trend) components associated with each of the elements of the vector time series are first removed by recursive, fixed interval smoothing based on generalised random walk (GRW) models; while the vector of perturbational residuals obtained from this "detrending" step is then modelled as a vector AR or ARMA process.

Book ChapterDOI
01 Jun 1988
TL;DR: A methodology for the numerical solution of differential equations and integral equations which furnishes computer generated bounds of high quality is developed and furnishes the basis for fixed point iteration techniques which deliver the solution bounds.
Abstract: A methodology for the numerical solution of differential equations and integral equations which furnishes computer generated bounds of high quality is developed. Functions and operators on functions are implemented by means of computer representable counterparts, the latter being the constituents of ultra-arithmetic. An interval ultra-arithmetic is also developed. This furnishes the basis for fixed point iteration techniques which deliver the solution bounds. High quality of these bounds results from the method of iterative residual correction in a function space.

13 Apr 1988
TL;DR: A method has been introduced which would result in improved distribution of basis weight over the sheet and can be viewed as non-trivial but straightforward application of the theory of Bayes-based LQG adaptive controllers.
Abstract: Basis weight is one of the most important characteristics determining the quality of paper. Adaptive control of the average basis weight in machine direction has been applied successfully on several Czechoslovak machines. A method has been introduced which would result in improved distribution of basis weight over the sheet. The method can be viewed as non-trivial but straightforward application of the theory of Bayes-based LQG adaptive controllers.

Journal ArticleDOI
TL;DR: In this article, it was shown that a barrelled space has a dense infinite-codimensional vector subspace, provided that it does not have its strongest locally convex topology.
Abstract: This note presents a structure theorem for locally convex barrelled spaces. It is shown that, corresponding to any Hamel basis, there is a natural splitting of a barrelled space into a topological sum of two vector subspaces, one with its strongest locally convex topology. This yields a simple proof that a barrelled space has a dense infinite-codimensional vector subspace, provided that it does not have its strongest locally convex topology. Some further results and examples discuss the size of the codimension of a dense vector subspace.

Journal ArticleDOI
TL;DR: The balanced expansion method does not introduce any of the variational instabilities that have plagued the early Dirac-Fock basis expansion calculations as mentioned in this paper, and as expansion size increases, the total energy smoothly converges toward the numerical Dirac−Fock limit from above.
Abstract: Dirac–Fock balanced Gaussian basis calculations that employ an extended nucleus model (see Ref. 11) have been performed on Be and Ne atoms by systematically enlarging the basis set. As expansion size increases, the total energy smoothly converges toward the numerical Dirac–Fock limit from above. The balanced expansion method does not introduce any of the variational instabilities that have plagued the early Dirac–Fock basis expansion calculations.

Journal ArticleDOI
TL;DR: In this article, a comparison of three variational principles commonly used in scattering problems, namely those due to Kohn (KVP), Schwinger (SVP), and Newton (NVP), is conducted by computing K-matrix elements for elastic scattering from nine different interaction potentials.
Abstract: Comparisons of three variational principles commonly used in scattering problems, namely those due to Kohn (KVP), Schwinger (SVP), and Newton (NVP), are presented. These comparisons are conducted by computing K‐matrix elements for elastic scattering from nine different interaction potentials. We represent the KVP trial functions as expansions containing two non‐L2 terms that represent the asymptotic free wave, and a set of L2 functions, while the SVP and the NVP trial functions are expansions containing only the L2 terms. Three different sets of L2 functions are used to examine the effect of changing the basis on the convergence characteristics of the three methods. We find that the rates of convergence for the Kohn, Schwinger, and Newton methods are strongly dependent on the nature of the potential and the basis set used. We also find that purely repulsive potentials are, in general, easier to converge than purely attractive potentials.

Journal ArticleDOI
TL;DR: In this article, the authors compared the NMC and NHF SCF results for Cu2 with calculations employing Gaussian basis set expansions and found that nearly all previous SCF calculations have underestimated the bond length by about the same amount (0.03 A) as that attributed to the unlinked cluster and relativistic corrections.
Abstract: NHF and NMCSCF results for Cu2 are compared with calculations employing basis set expansions. We find that nearly all previous SCF calculations using Gaussian basis sets have underestimated the bond length by about the same amount (0.03 A) as that attributed to the unlinked cluster and relativistic corrections. The error is shown to be due to deficiencies in the 3d primitive set which yield sizable basis set superposition errors.

Proceedings ArticleDOI
09 May 1988
TL;DR: In this article, a phase space representation for radiation from a space-time truncated aperture distribution has been implemented to charting the near to far zone evolution of the emitted pulse.
Abstract: In the analysis of focused radiation from large aperture systems, and especially for the generation of "bullet-like" strongly collimated fields under transient conditions, it is suggestive to employ field representations in terms of focused basis elements. Possible basis functions in the frequency domain include Gaussians and Hermite or Laguerre Gaussians, and, directly in the time domain, complex source pulsed beams and focus wave modes. Some difficulties concerning the excitability of the focus wave modes will be discussed. The basis elements are next embedded in a discretized phase space spanning the space-time and spatial-temporal frequency domains to yield a rigorous field representation. Using space-time Gaussians, the phase space representation has been implemented for radiation from a space-time truncated aperture distribution. The results reveal the effectiveness of this approach to charting the near to far zone evolution of the emitted pulse.