Factoring Matrices into the Product of Circulant and Diagonal Matrices

A generic matrix \(A\in \,\mathbb {C}^{n \times n}\) is shown to be the product of circulant and diagonal matrices with the number of factors being \(2n-1\) at most. The demonstration is constructive, relying on first factoring matrix subspaces equivalent to polynomials in a permutation matrix over diagonal matrices into linear factors. For the linear factors, the sum of two scaled permutations is factored into the product of a circulant matrix and two diagonal matrices. Extending the monomial group, both low degree and sparse polynomials in a permutation matrix over diagonal matrices, together with their permutation equivalences, constitute a fundamental sparse matrix structure. Matrix analysis gets largely done polynomially, in terms of permutations only.     

Alignment and switching of AFLC materials using various boundary conditions

We have systematically studied the quality of bookshelf alignment and electro‐optic characteristics of two antiferroelectric liquid crystal materials in cells with various boundary conditions. The electro‐optic characteristics of the materials studied depend strongly on both the liquid crystal materials and the boundary conditions at the supporting substrates. We have compared a number of observations in these cells: the tendency to form AFLC domains in the virgin state and after switching; the surface electroclinic effect (SEC effect); the transmission–voltage characteristics (TV) when driven with triangular‐ and square‐wave voltages at various frequencies; the threshold field and the conditions for relaxation to the AFLC state. The set of samples includes specially designed and manufactured test cells with different polyimides as alignment layers, treated with varying rubbing strengths. We discuss the significance of various factors and show the importance of simultaneously optimizing both materials and cell parameters for AFLC applications.     

Mikheyev–Smirnov–Wolfenstein Effect¶for Linear Electron Density

When the electron density is a linear function of distance, it is known that the MSW equations for two neutrino species can be solved in terms of known functions. It is shown here that more generally, for any number of neutrino species, these MSW equations can be solved exactly in terms of single integrals. While these integrals cannot be expressed in terms of known functions, some of their simple properties are obtained. Application to the solar neutrino problem is briefly discussed. Received: 3 April 2000 / Accepted: 10 April 2000     

Regularization Tools version 4.0 for Matlab 7.3

This communication describes version 4.0 of Regularization Tools, a Matlab package for analysis and solution of discrete ill-posed problems. The new version allows for under-determined problems, and it is expanded with several new iterative methods, as well as new test problems and new parameter-choice methods.      

Evaluating Nearly Singular Multinormal Expectations with Application to Wave Distributions

The numerical computation of expectations for (nearly) singular multivariate normal distribution is a difficult problem, which frequently occurs in widely varying statistical contexts. In this article we discuss several strategies to improve the algorithm proposed by Genz and Kwong (2000) when either a moderate accuracy is requested, the correlation structure is strong, and, most importantly, the dimension of the integral is large. Test results for typical problems show an average speedup of 10 using the modified algorithm, but even more is gained as the dimension of the problem increases. We apply the modified algorithm to compute long-run distributions of Gaussian wave characteristics, a difficult problem where previous algorithms fail to compute accurate values in reasonable time. AMS 2000 Subject Classification 65C60, 65D15, 68W25     

Analytical quality specification for measurements reported on an ordinal scale

A model for evaluation of analytical quality specifications for measurements performed on an ordinal scale is described. It is based on the assumption that the quantity, in principle, can be measured on a ratio (or difference) scale by other methods, but is performed by a cheap, simple and rapid method as 0 or 1 (minus or plus). It is not possible to estimate mean and standard deviations from an ordinal scale, so the measurements of 0 or 1 are characterized by the fraction of measured values of 1 for a certain concentration. For a series of increasing (known) concentrations of the quantity, the fractions of measured values of 1 are plotted on a probit scale. For a pregnancy test with measurements on an ordinal scale – measuring urine-human chorionic gonadotropin as minus or plus – allowable fractions of measured values of 1 are defined for the 'true' concentrations of 3 and 25 U/l based on clinical goals for early detection of pregnancy.     

Conservative solution of the Fokker–Planck equation for stochastic chemical reactions

In this paper we discuss multigrid methods for ill-conditioned symmetric positive definite block Toeplitz matrices. Our block Toeplitz systems are general in the sense that the individual blocks are not necessarily Toeplitz, but we restrict our attention to blocks of small size. We investigate how transfer operators for prolongation and restriction have to be chosen such that our multigrid algorithms converge quickly. We point out why these transfer operators can be understood as block matrices as well and how they relate to the zeroes of the generating matrix function. We explain how our new algorithms can also be combined efficiently with the use of a natural coarse grid operator. We clearly identify a class of ill-conditioned block Toeplitz matrices for which our algorithmic ideas are suitable. In the final section we present an outlook to well-conditioned block Toeplitz systems and to problems of vector Laplace type. In the latter case the small size blocks can be interpreted as degrees of freedom associated with a node. A large number of numerical experiments throughout the article confirms convincingly that our multigrid solvers lead to optimal order convergence. AMS subject classification (2000) 65N55, 65F10