Computing Complex Airy Functions by Numerical Quadrature

Integral representations are considered of solutions of the Airy differential equation w –zw=0 for computing Airy functions for complex values of z. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss–Laguerre quadrature; this approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.     

A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations

After studying Gaussian type quadrature formulae with mixed boundary conditions, we suggest a fast algorithm for computing their nodes and weights. It is shown that the latter are computed in the same manner as in the theory of the classical Gauss quadrature formulae. In fact, all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. Hence, we can adapt existing procedures for generating such quadrature formulae. Comparative results with various methods now in use are given. In the second part of this paper, new algorithms for spectral approximations for second-order elliptic problems are derived. The key to the efficiency of our algorithms is to find an appropriate spectral approximation by using the most accurate quadrature formula, which takes the boundary conditions into account in such a way that the resulting discrete system has a diagonal mass matrix. Hence, our algorithms can be used to introduce explicit resolutions for the time-dependent problems. This is the so-called lumped mass method. The performance of the approach is illustrated with several numerical examples in one and two space dimensions.

     

On the Wigner-Witmer correlation rules for a homonuclear diatomic molecule with the like atoms in identical atomic states

The problem of building the manifold of electronic states of a homonuclear diatomic molecule formed from two like atoms in identical atomic states is readdressed. A conceptually simple approach is presented by invoking a model and all the standard results originally obtained by Wigner and Witmer are reproduced.     

An evaluation of the radial part of numerical integration commonly used in DFT

Recently, Gill and Chien introduced a new radial quadrature for multiexponential integrands (MultiExp grid) to deal with the radial part of the numerical integration. In this article, the MultiExp grid is studied and used to integrate the charge density. The MultiExp grid, along with an optimal pruning scheme, performed very well both in terms of accuracy and efficiency compared to other radial mappings commonly used in Density Functional Theory.     

Four-dimensional lattice rules generated by skew-circulant matrices

We introduce the class of skew-circulant lattice rules. These are -dimensional lattice rules that may be generated by the rows of an skew-circulant matrix. (This is a minor variant of the familiar circulant matrix.) We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree . We describe some of the results of computer-based searches for optimal four-dimensional skew-circulant rules. Besides determining optimal rules for we have constructed an infinite sequence of rules that has a limit rho index of . This index is an efficiency measure, which cannot exceed 1, and is inversely proportional to the abscissa count.

     

A Framework for Evaluating Knowledge-Based Interestingness of Association Rules

In Knowledge Discovery in Databases (KDD)/Data Mining literature, interestingness measures are used to rank rules according to the interest a particular rule is expected to evoke. In this paper, we introduce an aspect of subjective interestingness called item-relatedness. Relatedness is a consequence of relationships that exist between items in a domain. Association rules containing unrelated or weakly related items are interesting since the co-occurrence of such items is unexpected. Item-Relatedness helps in ranking association rules on the basis of one kind of subjective unexpectedness. We identify three types of item-relatedness – captured in the structure of a fuzzy taxonomy (an extension of the classical concept hierarchy tree). An item-relatedness measure for describing relatedness between two items is developed by combining these three types. Efficacy of this measure is illustrated with the help of a sample taxonomy. We discuss three mechanisms for extending this measure from a two-item set to an association rule consisting of a set of more than two items. These mechanisms utilize the relatedness of item-pairs and other aspects of an association rule, namely its structure, distribution of items and item-pairs. We compare our approach with another method from recent literature.     

Hermiticity of Quantum Observables Versus Commutation Relations

In order to obtain sum rules and spectral representations the Hermiticity property , A = A, of observables is used. It is shown that for certain and the property turns out to be inconsistent with the commutation relations that contain A. The known Schwinger paradox is explained by this inconsistency.     

Radial quadrature for multiexponential integrands

We introduce a Gaussian quadrature, based on the polynomials that are orthogonal with respect to the weight function ln(2)x on the interval [0, 1], which is suitable for the evaluation of radial integrals. The quadrature is exact if the non-Jacobian part of the integrand is a linear combination of a geometric sequence of exponential functions. We find that the new scheme is a useful alternative to existing approaches, particularly for integrands that exhibit multiexponential behavior.     

Symbolic Computation of Newton Sum Rules for the Zeros of Polynomial Eigenfunctions of Linear Differential Operators

A symbolic algorithm based on the generalized Lucas polynomials of first kind is used in order to compute the Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators with polynomial coefficients.