Showing papers on "Basis (linear algebra) published in 1991"

A complete basis set model chemistry. III. The complete basis set‐quadratic configuration interaction family of methods

Abstract: The major source of error in most ab initio calculations of molecular energies is the truncation of the one‐electron basis set. A family of complete basis set (CBS) quadratic CI (QCI) model chemistries is defined to include corrections for basis set truncation errors. These models use basis sets ranging from the small 6‐31 G°° double zeta plus polarization (DZ+P) size basis set to the very large (14s9p4d2f,6s3p1d)/[6s6p3d2f,4s2p1d] atomic pair natural orbital basis set. When the calculated energies are compared with the experimental energies of the first‐row atoms and ions and the first‐row diatomics and hydrides H2, LiH, Li2, CH4, NH3, H2O, HF, LiF, N2, CO, NO, O2, and F2, two very promising new model chemistries emerge. The first is of comparable accuracy, but more than ten times the speed of the G1 model of Pople and co‐workers. The second is less than one‐tenth the speed of the G1 model, but reduces the root‐mean‐square (rms) errors in ionization potentials (IPs), electron affinities (EAs), and D0’s to 0.033 and 0.013 eV, and 0.53 kcal/mol per bond, respectively.

Visualizing vector field topology in fluid flows

Abstract: Methods for automating the analysis and display of vector field topology in general, and flow topology in particular, are described. By using techniques to extract and visualize topological information, it is possible to combine the simplicity of schematic depictions with the quantitative accuracy of curves and surfaces computed directly from the data. Two-dimensional vector field topology is discussed, covering critical points and time-dependent flows, to provide a basis for the examination of topology in three-dimensional separated flows. Surface topology and separation structures in three-dimensional flows are then addressed. The construction of representations of tangent surfaces that are accurate, as well as efficient to compute and display, is examined, covering tessellation, clipping, and refinement. Locating, characterizing, and displaying three-dimensional critical points are considered. >

Motion planning for controllable systems without drift

Abstract: A general strategy for solving the motion planning problem for real analytic, controllable systems without drift is proposed. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added. Using formal calculations with a product expansion relative to P. Hall basis, another control is produced that achieves the desired result on the formal level. This provides an exact solution of the original problem if the given system is nilpotent. For a general system, an iterative algorithm is derived that converges very quickly to a solution. For nonnilpotent systems which are feedback nilpotentizable, the algorithm, in cascade with a precompensator, produces an exact solution. Results of simulations which illustrate the effectiveness of the procedure are presented. >

Pattern representation model training apparatus

Abstract: Disclosed is an Hidden Markov Model (HMM) training apparatus in which a capacity for discriminating between models is taken into consideration so as to allow a high level of recognition accuracy to be obtained. A probability of a vector sequence appearing from HMMs is computed with respect to an input vector and continuous mixture density HMMs. Through this computation, the nearest different-category HMM, with which the maximum probability is obtained and which belongs to a category different from that of a training vector sequence of a known category, is selected. The respective central vectors of continuous densities constituting the output probability densities of the same-category HMM belonging to the same category as that of the training vector sequence and the nearest different-category HMM are moved on the basis of the vector sequence.

Transformation groups in net spaces

Abstract: We consider formal groups of transformations on the space of differential and net (finite-difference) variables. We show that preservation of meaning of difference derivatives under transformations necessarily leads to Lie-Backlund group. We derive formulas for extension to net variables and formulate criteria for preservation of uniformity and invariance of differences of the network and a test for the invariance of difference equations. With the help of formal Newton series we construct the ideal of the algebra of all Lie-Backlund operators on a uniform network which is used to derive tests for the conservatism of difference equations on the basis of a discrete analog of Noether's identity.

Atomic Natural Orbital (ANO) Basis Sets for Quantum Chemical Calculations

Abstract: Publisher Summary This chapter introduces an approach to basis set design— that is, to use physically motivated atomic orbitals as basis functions for molecular calculations. The fact that the AOs are expanded as fixed contractions of Gaussian functions is a physically irrelevant computational device. For correlated calculations, the natural orbitals derived from correlated atomic wave functions prove to be a compact, computationally efficient, and physically meaningful set of basis functions. The efficient evaluation of AO integrals over such basis sets imposes certain computational requirements on an integral code. The chapter discusses how these requirements are implemented in evaluation of two-electron integrals over a generally contracted Gaussian basis set in the molecule integral program. The chapter also discusses how ANO basis sets are effectively able to exhaust the capabilities of the underlying Gaussian expansion basis— that is, to minimize the contraction error. Finally, the chapter describes several applications of quantum chemistry to problems in which the use of ANO basis sets is significant in obtaining the required high accuracy.

A Hilbert space multi-reference coupled-cluster study of the H4 model system

Abstract: Employing the Hilbert space ansatz, a fully quadratic coupled-cluster method with a multidimensional reference space is applied to a DZP basis study of the model system, H4. The reference space is described by two to four configurations at the level of single and double excitations, and single and double excitation operators are included in the expansions for the cluster and wave operator through quadratic terms. The performance of quadratic MRCCSD is investigated for the ground and three excited states of the H4 system consisting of two stretched hydrogen molecules in a trapezoidal configuration where the degree of quasidegeneracy is varied from a nondegenerate situation to a completely degenerate one. Compared to full CI, in the highly degenerate region, the MRCCSD works quite well. In less degenerate regions, the accuracy is less satisfactory.

On Finding Primal- and Dual-Optimal Bases

Abstract: We show that if there exists a strongly polynomial time algorithm that finds a basis which is optimal for both the primal and the dual problems, given an optimal solution for one of the problems, then there exists a strongly polynomial algorithm for the general linear programming problem. On the other hand, we give a strongly polynomial time algorithm that finds such a basis, given any pair of optimal solutions (not necessarily basic) for the primal and the dual problems. Such an algorithm is needed when one is using an interior point method and is interested in finding a basis which is both primal- and dual-optimal. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Reduced basis approximations of structural displacements for optimaldesign

Abstract: A new solution procedure for improved approximate reanalyses of structures, using results of a single precise analysis, is presented. The proposed procedure is based on combining the computed terms of a series expansion, used as high-quality basis vectors, with coefficients of a reduced basis expression. The latter coefficients can readily be determined for each trial design by various criteria. The results are measured by the errors in satisfying the analysis equations and are compared in terms of the quality of the approximations. The proposed approach is suitable for various types of design variables and can be used with a general finite element model. A general reanalysis procedure is introduced and the physical significance of first-order approximations is demonstrated. Several numerical examples illustrate the effectiveness of the solution process. It is shown that high-quality approximations of displacements can be obtained with a small computational effort for very large changes in the design variables.

Overall properties of heterogeneous material

Abstract: Summary Small-scale heterogeneity in crustal structure can be very complex and difficult to describe in detail and yet, at the same time, can be very important for the description of, for instance, tectonic stress and porosity. Statistical properties of such heterogeneity can be derived from the properties of waves of relatively large wavelength as they propagate through the crust. The differences between measured wavelengths and attenuation coefficients and those of compact rock and the variations of these quantities in space and time provide, in principle, a means of determining quantities like the density and orientations of microfractures and the nature of crack infill material. The theoretical basis for inferences of this kind is the concept of the ‘effective’ or ‘equivalent’ material based on an averaging process taken over the microstructure. A number of methods have been used to calculate the properties of the effective medium, several of which are described here.

Three-dimensional impedance imaging processes

Abstract: A method for obtaining sets of current patterns for three-dimensionally imaging the interior of a body having an internal resistivity using electrical impedance tomography comprises providing an array of electrodes arranged in a plurality of groups for an impedance imaging system. A linearly independent set of current patterns is also established for forming a basis for each group. A constant pattern is then adjoined to each basis for forming an augmented basis for each group. A tensor product is then taken of the augmented basis for forming a tensor product basis. Finally, the constant pattern from the tensor product basis is removed in order to establish a basis of current patterns for being applied to the array of electrodes.

Dirac-Fock-Breit self-consistent-field method: Gaussian basis-set calculations on many-electron atoms

Abstract: The self-consistent-field treatment of the frequency-independent Breit interaction is reviewed with applications to many-electron atoms. The implementation of the matrix Dirac-Fock-Breit self-consistent-field procedure is presented for Gaussian-type basis sets that show no near-linear dependency problem. The matrix Dirac-Fock-Breit procedure has the advantage over the finite-difference approach that it does not complicate the self-consistent-field procedure in basis-set expansion calculations. Basis sets of even- and well-tempered Gaussian functions were used to expand the large and small components of Dirac four-spinors. Expressions are derived for evaluating the matrix elements of the Dirac-Fock-Breit equations. Calculations done on rare-gas atoms He, Ne, Ar, Kr, and Xe and alkaline-earth metals Be, Mg, Ca, and Sr are presented.

Vector quantizing apparatus and speech analysis-synthesis system using the apparatus

Abstract: A vector quantizing apparatus having a general vector quantization circuit, and a storage means for storing at least one frame of data as the result of comparison by a matching circuit is provided. Further, provided are a speech analysis-synthesis system having a spectral envelope generator for generating a spectral envelope which is so smooth that excessive beating is avoided, a spectral envelope vector converter for sampling the spectral envelope at equal intervals on mel-scale, a vector quantizer for quantizing vectors, and a spectral envelope reconstructor for reconstructing the spectral envelope by interpolation on the basis of combined parabolas.

Progress of basis optimization techniques in variational calculation of quantum reactive scattering

Abstract: This paper describes several efficient basis optimization methods that we have developed in the application of S‐matrix Kohn variational method to quantum reactive scattering. Specifically, we employ a minimum‐K body‐fixed representation combined with the use of quasiadiabatic basis functions for the expansion of the full reactive scattering wave function. This new basis function approach significantly reduces the size of the ‘‘larger’’ matrix of the final linear algebraic equation in the calculation of reaction cross sections. The accuracy of the calculation can be easily controlled by systematically increasing or decreasing the values of two parameters Kmax and α, and convergence to the full basis set results can be reached. Numerical test calculations are carried out for the 3D H+H2 reaction for the total angular momentum J=10 and for the 3D F+H2 reaction for J=0, 1, and 2. These calculations demonstrate that our basis optimization approach is very efficient for computing reaction cross sections. Since...

Resolving redundant manipulator joint rates and identifying special arm configurations using Jacobian null-space bases

Abstract: Joint rate solutions for redundant manipulators are formulated in terms of a particular solution and a homogeneous solution formed using a basis for the Jacobian null space Solutions for several local optimization objectives are formulated in terms of Jacobian null-space bases A method based on a decomposition of screw coordinates is presented for finding null-space bases and particular solutions The screw decomposition is applied in the derivation of the analytical expressions for the null-space bases and in the identification of special configurations for a seven-revolute manipulator The analysis is then extended to multiple-arm systems with common degrees of freedom, and is applied to an analytical example involving industrial manipulators mounted on a common platform >

Approximation theorems for discrete-time systems

Abstract: It is shown that the members of an important class of input-output maps G can be uniformly approximated arbitrarily well by finite sums of a certain form, involving a simple linear operator, row vectors, and a mapping from R to R. This can be used, for example, as a basis for adaptive filtering or for the adaptive identification of G's. >

A Model of Stream Channel Adjustment: Assessment of Rubey's Hypothesis

Abstract: W. W. Rubey proposed a concept of stream channel development with channel gradient and channel form as dependent variables that mutually adjust to conditions of sediment load, sediment size, and water discharge. A similar channel development model is formulated using multiple equations and tested against a data set for 252 stream sites in the Missouri River Basin, USA. The results strongly support mutual adjustment among the gradient, form, width, and depth aspects of the stream channel system. The relation between channel form and grain size in the bed is shown to have opposite signs in the structural and reduced forms of the model. This provides the basis for an explanation of a puzzling aspect of Rubey's conception of the adjustment process. The results suggest that channel form is more dependent on the process of mutual adjustment of morphologic elements than is channel gradient.

Ab initio SCF calculations on [V10O28]6−: A benchmark for the classical calculation and processing of molecular integrals on large Gaussian basis sets

Abstract: A vector efficient implementation of the McMurchie and Davidson algorithm for the calculation of one- and two-electron molecular integrals is presented, as available in the Cray version of the ASTERIX program system. The implementation and performance of a vector-oriented strategy for the generation and processing of the P supermatrix is also discussed

Spectral expansion analysis in electrical impedance tomography

Abstract: The EIT image domain is expanded in a complete set of orthogonal basis images. The number of basis images is given by the number of independent measurements (104 for 16 electrodes). They are ordered according to their sensitivity to data noise, with centrally located features and higher spatial frequency components appearing towards the higher terms in the series. This indicates that the resolution can be improved at the expense of degrading signal to noise in the reconstructed image. The technique offers a flexible way of assembling a reconstruction protocol. A specific reconstruction algorithm similar to the algebraic reconstruction technique (ART) is analysed to show its regularization properties.

Planning controlled clinical trials on the basis of descriptive data analysis.

Abstract: In controlled clinical trials the problem of multiplicity of desired inferential statements finds attention at an increasing rate. In this paper the previously proposed concept of Descriptive Data Analysis (DDA), situated between Confirmatory and Exploratory Data Analysis, is applied to the planning aspects of controlled trials for which the problem of multiplicity exists. The (non-Bayesian) DDA planning concept should provide the investigator with tools to draw final conclusions from data of several variables possibly observed at several time points in possibly several groups of subjects by combining his pre-trial medical experience with descriptive inferential statements (confidence intervals and test results) at nominal significance levels. DDA also provides for confirmatory statements concerning individual null hypotheses and partially global hypotheses.

Weakly null sequences with upper ℓp-estimates

Abstract: A Banach space X has property (5) if every weakly null sequence in BaX, the unit ball of X, has a subsequence which is C-dominated by the unit vector basis of Co for some constant C < 00. In [11] it was shown that if X has property (5), then the constant C can be chosen to be independent of the particular weakly null sequence in B a X. Here we generalize this result to the case of upper £p-estimates.

Adaptive control system

Abstract: In an adaptive control system, an input of an object to be controlled is varied by a small value, and a resultant output is detected. The small value is changed on the basis of the difference between the output and an aimed value, and trials are repeated. The small value is determined on the basis of a qualitative model which is composed of a boundary function including the input and boundary parameters of the object. If the trend in the difference between the output and the aimed value is to make it larger, the qualitative model is changed so that the trend is to make it smaller.