On the Sauer-Xu formula for the error in multivariate polynomial interpolation

Use of a new notion of multivariate divided difference leads to a short proof of a formula by Sauer and Xu for the error in interpolation, by polynomials of total degree in variables, at a `correct' point set.

     

多项式带余除法的差商表示

利用差商和多项式插值理论,给出了两个多项式相除的商式和余式的差商表达式.     

Evaluating higher derivative tensors by forward propagation of univariate Taylor series

This article considers the problem of evaluating all pure and mixed partial derivatives of some vector function defined by an evaluation procedure. The natural approach to evaluating derivative tensors might appear to be their recursive calculation in the usual forward mode of computational differentiation. However, with the approach presented in this article, much simpler data access patterns and similar or lower computational counts can be achieved through propagating a family of univariate Taylor series of a suitable degree. It is applicable for arbitrary orders of derivatives. Also it is possible to calculate derivatives only in some directions instead of the full derivative tensor. Explicit formulas for all tensor entries as well as estimates for the corresponding computational complexities are given.

     

Mathematical proof of closed form expressions for finite difference approximations based on Taylor series

Taylor series based finite difference approximations of derivatives of a function have already been presented in closed forms, with explicit formulas for their coefficients. However, those formulas were not derived mathematically and were based on observation of numerical results. In this paper, we provide a mathematical proof of those formulas by deriving them mathematically from the Taylor series.     

Intertwining unisolvent arrays for multivariate Lagrange interpolation

Generalizing a classical idea of Biermann, we study a way of constructing a unisolvent array for Lagrange interpolation in C^n+m out of two suitably ordered unisolvent arrays respectively in Cⁿ and C^m. For this new array, important objects of Lagrange interpolation theory (fundamental Lagrange polynomials, Newton polynomials, divided difference operator, vandermondian, etc.) are computed. AMS subject classification 41A05, 41A63     

Breaking the limits: The Taylor series method

This paper discusses several examples of ordinary differential equation (ODE) applications that are difficult to solve numerically using conventional techniques, but which can be solved successfully using the Taylor series method. These results are hard to obtain using other methods such as Runge-Kutta or similar schemes; indeed, in some cases these other schemes are not able to solve such systems at all. In particular, we explore the use of the high-precision arithmetic in the Taylor series method for numerically integrating ODEs. We show how to compute the partial derivatives, how to propagate sets of initial conditions, and, finally, how to achieve the Brouwer’s Law limit in the propagation of errors in long-time simulations. The TIDES software that we use for this work is freely available from a website.     

Efficient algorithms for generating truncated multivariate normal distributions

Sampling from a truncated multivariate normal distribution (TMVND) constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative data augmentation (DA) algorithm and a non-iterative inverse Bayes formulae (IBF) sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints. By creating a Bayesian incomplete-data structure, the posterior step of the DA algorithm directly generates random vector draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical software packages such as S-PLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative to Gibbs sampling and eliminates problems with convergence and possible slow convergence due to the high correlation between components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than the Gibbs sampler and the accept-reject algorithm.     

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

     

An algorithm for determining the approximation orders of multivariate periodic spline spaces

We give an algorithm which computes the approximation order of spaces of periodic piece-wise polynomial functions, given the degree, the smoothness and tesselation. The algorithm consists of two steps. The first gives an upper bound and the second a lower bound on the approximation order. In all known cases the two bounds coincide.     

A fast algorithm for multivariate Hermite interpolation

Multivariate Hermite interpolation is widely applied in many fields, such as finite element construction, inverse engineering, CAD etc.. For arbitrarily given Hermite interpolation conditions, the typical method is to compute the vanishing ideal I （the set of polynomials satisfying all the homogeneous interpolation conditions are zero） and then use a complete residue system modulo I as the interpolation basis. Thus the interpolation problem can be converted into solving a linear equation system. A generic algorithm was presented in [18], which is a generalization of BM algorithm [22] and the complexity is O（τ^3） where r represents the number of the interpolation conditions. In this paper we derive a method to obtain the residue system directly from the relative position of the points and the corresponding derivative conditions （presented by lower sets） and then use fast GEPP to solve the linear system with O（（τ ＋ 3）τ^2） operations, where τ is the displacement-rank of the coefficient matrix. In the best case τ = 1 and in the worst case τ = [τ/n], where n is the number of variables.     

Asymptotic formulae for multivariate Kantorovich type generalized sampling series

In this paper an asymptotic formula of Voronovskaja type for a multivariate extension of the Kantorovich generalized sampling series is given. Moreover a quantitative version in terms of some moduli of smoothness is established. Finally some particular examples of kernels are discussed, as the Bochner-Riesz kernel and the multivariate splines.     

The use of the tetrachoric series for evaluating multivariate normal probabilities

The tetrachoric series is a technique for evaluating multivariate normal probabilities frequently cited in the statistical literature. In this paper we have examined the convergence properties of the tetrachoric series and have established the following. For orthant probabilities, the tetrachoric series converges if $\text{|;?}{}_{\mathrm{ij}}\text{|; <}\text{1}\text{(k ? 1)}$, 1 ≤ i < j ≤ k, where ?_ij are the correlation coefficients of a k-variate normal distribution. The tetrachoric series for orthant probabilities diverges whenever k is even and $\text{?}{}_{\mathrm{ij}}\text{>}\text{1}\text{(k ? 1)}$ or k is odd and $\text{?}{}_{\mathrm{ij}}\text{>}\text{1}\text{(k ? 2)}$, 1 ≤ i < j ≤ k. Other specific results concerning the convergence or divergence of this series are also given. The principal point is that the assertion that the tetrachoric series converges for all k ≥ 2 and all ?_ij such that the correlation matrix is positive definite is false.     

A remainder formula of numerical differentiation for the generalized Lagrange interpolation

Through exploiting the generalized Lagrange interpolation for continuous piecewise smooth functions, a remainder term of numerical differentiation is given. Also, the error estimates for the numerical differentiation formula are presented.     

A new algorithm for computing the multivariate Faà di Bruno’s formula

A new algorithm for computing the multivariate Faà di Bruno’s formula is provided. We use a symbolic approach based on the classical umbral calculus that turns the computation of the multivariate Faà di Bruno’s formula into a suitable multinomial expansion. We propose a MAPLE procedure whose computational times are faster compared with the ones existing in the literature. Some illustrative applications are also provided.     

Lagrange‐type basis for multivariate uniform integrable tensor‐product Birkhoff interpolation

Birkhoff interpolation is the most general interpolation scheme. We study the Lagrange‐type basis for uniform integrable tensor‐product Birkhoff interpolation. We prove that the Lagrange‐type basis of multivariate uniform tensor‐product Birkhoff interpolation can be obtained by multiplying corresponding univariate Lagrange‐type basis when the integrable condition is satisfied. This leads to less computational complexity, which drops to from .     

Multiparameter estimation for some multivariate discrete distributions with possibly dependent components

Summary In multiparameter estimation for multivariate discrete distributions with infinite support, inadmissibility problems in situations where the multivariate probability distribution function isnot a product of the one-dimensional marginal probability distribution functions have previously been unexplored. This paper examines the inadmissibility problem in some of these situations. Special attention is given to estimating the mean of a negative multinomial distribution. In estimating the mean vector, certain Clevenson-Zidek type estimators are shown to be uniformly better than the usual estimator under a large class of generally scaled squared loss functions. Some of the results are generalized to other multivariate discrete distributions and to situations where several independent negative multinomial distributions are considered.     

Time shifted aliasing error upper bounds for truncated sampling cardinal series

Time shifted aliasing error upper bound extremals for the sampling reconstruction procedure are fully characterized. Sharp upper bounds are found on the aliasing error of truncated cardinal series and the corresponding extremals are described for entire functions from certain specific Lp, p>1, classes. Analogous results are obtained in multidimensional regular sampling. Truncation error analysis is provided in all cases considered. Moreover, sharpness of bounding inequalities, convergence rates and various sufficient conditions are discussed.     

H-bases for polynomial interpolation and system solving

The H-basis concept allows, similarly to the Gröbner basis concept, a reformulation of nonlinear problems in terms of linear algebra. We exhibit parallels of the two concepts, show properties of H-bases, discuss their construction and uniqueness questions, and prove that n polynomials in n variables are, under mild conditions, already H-bases. We apply H-bases to the solution of polynomial systems by the eigenmethod and to multivariate interpolation.