Combinatorial data analysis: Confirmatory comparisons between sets of matrices

The task of assessing the similarity of pattern between the entries of two square matrices has been discussed extensively over the last decade, as a unifying strategy for approaching a variety of seemingly disparate statistical problems. As typically defined, the comparison depends on a measure of matrix correspondence, usually a normalized cross-product measure of some form, that is evaluated for relative size by the use of a reference distribution constructed through an equally likely permutation hypothesis defined at the level of the objects corresponding to the rows and columns of the two matrices. The extreme generality provided by this very simple framework subsumes a variety of different statistical problems, ranging from the study of spatial autocorrelation for variables observed over a set of geographic locations, to the topics of analysis of variance, the measurement of rank correlation, and confirmation techniques concerned with various conjectures of combinatorial structure that might be posited for an empirically determined measure of relationship between pairs of a given set of objects. The comparison strategies extant always assume that both matrices are fixed, and in those cases where one of the matrices codifies a given theoretical structure to be evaluated according to a second, this assumption can lead to substantial arbitrariness in how matrix similarity might be indexed, and thus, in how the comparison is implemented. As developed in this paper, exactly the same principles appropriate for use in the fixed comparison context can be extended to include matrices constructed through optimally weighted linear combinations of other sets of matrices. This generalization provides one mechanism for developing comparison strategies that allow assessment against very broad classes of matrices, which in turn serve to represent very general conjectures of possible combinatorial structure. This paper reviews some of these extensions in detail, with a particular emphasis on categorical and ordered categorical variables and whether they may reflect an empirically generated measure of object relationship.     

Dual gradient method for linearly constrained,strongly convex,separable mathematical programming problems

A new dual gradient method is given to solve linearly constrained, strongly convex, separable mathematical programming problems. The dual problem can be decomposed into one-dimensional problems whose solutions can be computed extremely easily. The dual objective function is shown to have a Lipschitz continuous gradient, and therefore a gradient-type algorithm can be used for solving the dual problem. The primal optimal solution can be obtained from the dual optimal solution in a straightforward way. Convergence proofs and computational results are given.     

The smallest point of a polytope

This note suggests new ways for calculating the point of smallest Euclidean norm in the convex hull of a given set of points inR ⁿ . It is shown that the problem can be formulated as a linear least-square problem with nonnegative variables or as a least-distance problem. Numerical experiments illustrate that the least-square problem is solved efficiently by the active set method. The advantage of the new approach lies in the solution of large sparse problems. In this case, the new formulation permits the use of row relaxation methods. In particular, the least-distance problem can be solved by Hildreth's method.     

Second-order numerical method for domain optimization problems

This paper is concerned with a second-order numerical method for shape optimization problems. The first variation and the second variation of the objective functional are derived. These variations are discretized by introducing a set of boundary-value problems in order to derive the second-order numerical method. The boundary-value problems are solved by the conventional finite-element method.The authors would like to express their thanks to Mr. T. Masanao, who was an undergraduate student, for his cooperation and comments. They also thank Professor Y. Sakawa of Osaka University for his encouragement.A part of this paper was presented at the IFIP Conference on Control of Boundaries and Stabilization, Clermont-Ferrand, France, 1988.     

Computational schemes for large-scale problems in extended linear-quadratic programming

Numerical approaches are developed for solving large-scale problems of extended linear-quadratic programming that exhibit Lagrangian separability in both primal and dual variables simultaneously. Such problems are kin to large-scale linear complementarity models as derived from applications of variational inequalities, and they arise from general models in multistage stochastic programming and discrete-time optimal control. Because their objective functions are merely piecewise linear-quadratic, due to the presence of penalty terms, they do not fit a conventional quadratic programming framework. They have potentially advantageous features, however, which so far have not been exploited in solution procedures. These features are laid out and analyzed for their computational potential. In particular, a new class of algorithms, called finite-envelope methods, is described that does take advantage of the structure. Such methods reduce the solution of a high-dimensional extended linear-quadratic program to that of a sequence of low-dimensional ordinary quadratic programs.This work was supported in part by grants AFOSR 87-0821 and AFOSR 89-0081 from the Air Force Office of Scientific Research.     

Family of Twelve Steps Exponential Fitting Symmetric Multistep Methods for the Numerical Solution of the Schrödinger Equation

In the present paper we present a family of twelve steps symmetric multistep methods. The explicit part of new family of methods is applied to the scattering problems of the radial Schrödinger equation. This application shows the efficiency of the new family of methods.     

Symmetric Eighth Algebraic Order Methods with Minimal Phase-Lag for the Numerical Solution of the Schrödinger Equation

In this paper some new eighth algebraic order symmetric eight-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Based on this formula, the calculation of free parameters is done in order the phase-lag to be minimal. The new methods have better stability properties than the classical one. Numerical illustrations on the radial Schrödinger equation indicate that the new method is more efficient than older ones.     

Reducibility and contractivity of Runge–Kutta methods revisited

The exact relation between a Cooper-like reducibility concept and the reducibilities introduced by Hundsdorfer, Spijker and by Dahlquist and Jeltsch is given. A shifted Runge–Kutta scheme and a transplanted differential equation is introduced in such a fashion that the input/output relation remains unchanged under these transformations. This gives a technique to prove stability and contractivity results. This is demonstrated on the example of contractivity disks. AMS subject classification (2000) 65L07     

The employment of relativistic adapted Gaussian basis sets in Douglas–Kroll–Hess scalar calculations with diatomic molecules

The prolapse-free relativistic adapted Gaussian basis sets (RAGBSs), developed by our research group on the basis of the four-component approach, are used for the first time in Douglas–Kroll–Hess 2nd order scalar relativistic calculations (DKH2) of simple diatomic molecules containing Hydrogen and the halogens from Fluorine up to Iodine: HX and X₂, where X = F, Cl, Br, and I. To this end, the RAGBSs were contracted with the general contraction scheme to triple-, quadruple-, and quintuple-zeta sets. Polarization functions were also added to the basis sets by optimization with the configuration interaction method including single and double excitations into the DKH2 environment, DKH2-CISD. The molecular properties were then calculated with the coupled cluster electronic correlation treatment and the DKH2 scalar relativistic method, DKH2-CCSD(T), and indicated that our RAGBSs should be contracted as quadruple-zeta basis sets. The results achieved with the DKH2-CCSD(T) calculations and the selected quadruple-zeta RAGBSs are able to reproduce the experimental data of equilibrium distances, dissociation energies, and harmonic vibrational frequencies with root-mean-square (rms) errors of 0.015 Å, 3.6 kcal mol⁻¹, and 21.7 cm⁻¹, respectively.