An Estimation of the Truncation Error for the Two-Channel Sampling Formulas

The aim of the article is to obtain an estimation for the truncation error in the two-channel sampling formulas. Since these formulas are expansions with respect to suitable Riesz bases in Paley-Wiener spaces, the truncation error will be estimated by using the hypercircle inequality in the Riesz bases setting. In so doing, the norm of an involved operator is calculated, and the remainder of the series of the absolute square sampling functions is estimated.     

The Zero-Removing Property and Lagrange-Type Interpolation Series

The classical Kramer sampling theorem, which provides a method for obtaining orthogonal sampling formulas, can be formulated in a more general nonorthogonal setting. In this setting, a challenging problem is to characterize the situations when the obtained nonorthogonal sampling formulas can be expressed as Lagrange-type interpolation series. In this article a necessary and sufficient condition is given in terms of the zero removing property. Roughly speaking, this property concerns the stability of the sampled functions on removing a finite number of their zeros.     

The construction of high order convergent look-ahead finite difference formulas for Zhang Neural Networks

ABSTRACT

Zhang Neural Networks rely on convergent 1-step ahead finite difference formulas of which very few are known. Those which are known have been constructed in ad-hoc ways and suffer from low truncation error orders. This paper develops a constructive method to find convergent look-ahead finite difference schemes of higher truncation error orders. The method consists of seeding the free variables of a linear system comprised of Taylor expansion coefficients followed by a minimization algorithm for the maximal magnitude root of the formula's characteristic polynomial. This helps us find new convergent 1-step ahead finite difference formulas of any truncation error order. Once a polynomial has been found with roots inside the complex unit circle and no repeated roots on it, the associated look-ahead ZNN discretization formula is convergent and can be used for solving any discretized ZNN based model. Our method recreates and validates the few known convergent formulas, all of which have truncation error orders at most 4. It also creates new convergent 1-step ahead difference formulas with truncation error orders 5 through 8.     

On truncation error bound for multidimensional sampling expansion laplace transform

The truncation error associated with a given sampling representation is defined as the difference between the signal and an approximating sumutilizing a finite number of terms. In this paper we give uniform bound for truncation error of bandlimited functions in the n dimensional Lebesgue space Lp(R^n) associated with multidimensional Shannon sampling representation.     

Time step selection for the numerical solution of boundary value problems for parabolic equations

An algorithm is proposed for selecting a time step for the numerical solution of boundary value problems for parabolic equations. The solution is found by applying unconditionally stable implicit schemes, while the time step is selected using the solution produced by an explicit scheme. Explicit computational formulas are based on truncation error estimation at a new time level. Numerical results for a model parabolic boundary value problem are presented, which demonstrate the performance of the time step selection algorithm.     

一阶导数的五点数值微分公式及外推算法

通过对四次Lagrange插值多项式求导推导出一阶导数的五点数值微分公式,其截断误差为O(h~4).利用Richardson外推原理得到该公式的外推算法,K次外推后,中间节点的数值精度提高到O(h~(2(k+2))),其它节点的精度提高到O(h~(k+4)).     

Numerical integration by means of adapted Euler summation formulas

Summary The purpose of the paper is the study of formulas and methods for numerical integration based on Euler summation formulas. Cubature formulas are developed from multidimensional generalizations. Estimates of the truncation error are given in adaptation to specific properties of the integrand.     

用加权平均方法构造新的隐式线性多步法公式

在已知的线性多步法公式中,用两个较适合的线性多步法进行加权平均就能构造出一系列新的隐式线性多步法公式,而且其中有些公式可能具有较好的性质,如稳定域增大.从而使得解刚性方程时,可以根据对稳定域与截断误差不同的需求来选择公式,以达到在适合的稳定域下,截断误差最小.经过数值试验验证,本文举出的实例中用加权平均方法构造出的有些新公式的稳定域大于原来两个公式任一个的稳定域,可应用于求解常微分方程初值问题的刚性问题.     

Empirically estimating error of integration by quasi-Monte Carlo method

The possibility of applying probability-theoretical methods to a deterministic procedure for estimating the error of evaluating multiple integrals by the quasi-Monte Carlo method is considered. The existing methods for estimating this error are nonconstructive. The well-known Koksma-Hlawka inequality contains the variations of the integrand as a constant. The problem of calculating these variations is more difficult than the initial one. Since the quasi-Monte Carlo method uses the arithmetic mean value of the integrand as an estimate of the integral, it is natural to expect that the distribution (in the number-theoretic sense) of the remainder in the approximate integration procedure obeys the normal law. However, there is an additional difficulty that, from the probability-theoretic point of view, quasi-random points are dependent, and numerically estimating the second moment of the remainder is thereby impeded. An approach to estimating the second moment of the error is proposed, which is based on the results of the theory of random cubature formulas obtained by the authors. Numerical examples are given, which show that the proposed error estimation method has great potential.     

Expansion Over a Rectangle of Real Functions in Bernoulli Polynomials and Applications

In this paper we generalize an expansion in Bernoulli polynomials for real functions possessing a sufficient number of derivatives. Starting from this expansion we obtain useful kernels, which are substantially different from Sard's for a wide class of linear functionals that includes the truncation error for cubature formulas.This revised version was published online in October 2005 with corrections to the Cover Date.     

基于泰勒级数展开的一点超前差分公式的推导

传统的数值微分公式有前向差分、后向差分和中心差分公式.所谓一点超前差分公式,就是后向差分公式在形式上"前移"一点来计算一阶导数的公式.该公式有效地弥补了传统差分公式的不足之处.不同于以前研究中使用拉格朗日公式来推导一点超前公式的做法,给出了基于泰勒级数展开的对该组公式及其截断误差的推导,从另一个角度验证了一点超前公式,使其更为完善.     

Ostrowski不等式的改进及其应用

通过对Ostrowski不等式的改进,扩大了Ostrowski积分公式的适用范围,将该积分公式应用于数值积分推广了经典的中点积分公式、梯形积分公式和Simpson积分公式,同时得到相应的最佳误差限,并给出了具体的数值应用.     

Error analysis for the computation of zeros of regular Coulomb wave function and its first derivative

In 1975 one of the coauthors, Ikebe, showed that the problem of computing the zeros of the regular Coulomb wave functions and their derivatives may be reformulated as the eigenvalue problem for infinite matrices. Approximation by truncation is justified but no error estimates are given there.

The class of eigenvalue problems studied there turns out to be subsumed in a more general problem studied by Ikebe et al. in 1993, where an extremely accurate asymptotic error estimate is shown.

In this paper, we apply this error formula to the former case to obtain error formulas in a closed, explicit form.

     

Truncation Error on Wavelet Sampling Expansions

We estimate the truncation error of sampling expansions on translationinvariant spaces, generated by integer translations of a single functionand on wavelet subspaces of L ₂(R). As a byproduct of themain result, we get the classical Jagerman's bound for Shannon's samplingexpansions. We also examine this error on certain wavelet sampling expansions.     

PERT工序时间估计的一些研究

本文首先从理论上分析了在工序时间服从β—分布的条件下，传统的ＰＥＲＴ公式的估计误差，并指出其中的方差估计误差范围比较大．其次本文介绍了国外提出的若干工序时间估计改进方法，并对其相互优劣比较结果进行了评述．最后本文就工序时间估计的一些基本问题进行了讨论．     

A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling

The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established.     

“中国家庭经济与生育研究”抽样调查方案及目标量估计方法

本文论述了“中国家庭经济与生育研究”全国抽样调查中用聚类方法对全国２５省（市、区）和农村发达与落后地区进行分层,并采用多阶多次分层抽样;给出了在此方案下各层次目标量的估计方法和估计公式的变换形式．     

On Lagrange-Type Interpolation Series and Analytic Kramer Kernels

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. A challenging problem is to characterize the situations when these sampling formulas can be written as Lagrange-type interpolation series. This article gives a necessary and sufficient condition to ensure that when the sampling formula is associated with an analytic Kramer kernel, then it can be expressed as a quasi Lagrange-type interpolation series; this latter form is a minor but significant modification of a Lagrange-type interpolation series. Finally, a link with the theory of de Branges spaces is established. Received: October 8, 2007. Revised: December 13, 2007.     

Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes

Abstract

Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Finally, the stochastic Taylor formulas based on coloured trees turn out to be a generalization of the deterministic Taylor formulas using plain trees as recommended by Butcher for the solutions of ordinary differential equations.     

Error bounds in the gap metric for dissipative balanced approximations

We derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation. We prove these results by working in the context of dissipative driving-variable systems, as in behavioral and state/signal systems theory. In such a framework no prior distinction is made between inputs and outputs. Dissipativity preserving balanced truncation of dissipative driving-variable systems is addressed and a gap metric error bound is obtained. Bounded real and positive real input–state–output systems are manifestations of a dissipative driving-variable system through particular decompositions of the signal space. Under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow.