The construction of cubature formulae for a family of integrals: A bifurcation problem

Cubature formulae of degree 11 with minimal numbers of knots for the integral $$\int\limits_{ - 1}^1 { \int\limits_{ - 1}^1 {(1 - x^2 )^\alpha } } (1 - y^2 )^\alpha f(x,y) dxdy \alpha > - 1$$ which are invariant under rotation over an angle π/2 are determined by a system of 18 nonlinear equations in 18 unknowns. We start with a known solution for this system for α=0. By varying α smoothly, the knots and weights of the cubature formula vary smoothly except in the singular solutions such as turning points and bifurcation points where new solutions branches arise. We use for this purpose the program AUTO. We obtain surprisingly many branches of cubature formulae.     

The construction of symmetric cubature formulas for the square and the triangle

The weights and nodes of a symmetric cubature formula are determined by solving a system of nonlinear equations. The number of equations and their structure are investigated for symmetric cubature formulas for the square and the triangle. A new cubature formula of degree 7 with 12 nodes is given for the triangle.     

On the numerical solution of non-linear equation characterizing minimal cubature formulae

A cubature formula with degree of exactness ≦2k?2 has at leastk(k+1)/2 knots. The existence of such minimal formulae is equivalent to the existence of solutions of a system of quadratic equations. Each solution of such a system generates in a uniquely determined way a minimal formula. For the square [?1, 1]² as domain of integration and for some weight-functions these systems have a simple form, but they seem to be hard to solve. In this note attention is drawn to these systems and to experiences made in order to obtain numerical solutions.     

Zur Charakterisierung und Berechnung von symmetrischen Kubaturformeln

The paper is concerned with the characterization and calculation of symmetric cubature formulae of degree 2k?1 for two-dimensional product-functionals. The number of knots of the cubature formulae satisfies the following relation: $$\frac{{k(k + 1)}}{2} + \left[ {\frac{k}{2}} \right] \leqslant r \leqslant \frac{{k(k + 1)}}{2} + \left[ {\frac{k}{2}} \right] + 1.$$ The systems of non-linear equations involved are either solved exactly or all solutions are computed with any precision using a program-package for symbolic and algebraic calculations.     

Blending methods for two classical integrals

A construction method for cubature formulae of an arbitrary degree ford-dimensional product-integrals,d≥2, will be presented. To get a high degree of exactness with a moderate number of nodes quadrature rules are blended in a suitable way. For product-integrals with Tschebycheff-weight-functions the corresponding cubature formulae are minimal or ‘minimal + 1’ in the cased=2. In higher dimensions the number of nodes of the constructed formulae is far beyond the number of nodes of other approaches known.     

New bivariate polynomial expansion with boundary data on the simplex

We introduce an extension to the two-dimensional simplex of the univariate two-point expansion formula for sufficiently smooth real functions introduced in [13]; it is a polynomial expansion with algebraic degree of exactness. This expansion is applied to obtain a new class of embedded boundary-type cubature formulae on the simplex.      

Minimal cubature formulae of degree 2k−1 for two classical functionals

Cubature formulae with the number of nodes equal to Möller's lower bound are rare. In this paper, the relation between real polynomial ideals and cubature formulae is used to construct such minimal formulae of arbitrary odd degree for two classical integrals. We found general expressions for bases of these ideals and closed formulae for almost all nodes. We proved that all nodes are inside the domain of integration.     

均方根嵌入式容积卡尔曼滤波

传统容积卡尔曼滤波(CKF)的基础是三阶球面-径向容积准则,该准则不仅要求计算n维超球体上的面积分,还需将容积准则与扩展高斯-拉盖尔准则配合使用,不易推导出高阶CKF滤波算法.此外,CKF推导所采用的三阶球面容积准则也存在缺陷,这极大地限制了CKF的滤波精度.为避免以上问题,本文基于嵌入式容积准则和均方根滤波技术,提出一种加性噪声环境下,用于非线性动态系统状态估计的全新容积卡尔曼滤波算法-三阶均方根嵌入式容积卡尔曼滤波(SICKF).SICKF具有滤波精度高、数值稳定性强等诸多优点,适用于动态目标跟踪、非线性系统控制等.仿真结果表明,SICKF的滤波精度显著优于传统的非线性滤波算法.     

The construction of cubature formulae by continuation

A cubature formulaQ is an approximation of ann-dimensional integralI. Q is exact for the space spanned by the polynomialsf ₁, ...,f _d if it verifies the system of equations: $$Q[f_i ] = I[f_i ] i = 1,...,d.$$ The unknowns are knots and weights of the cubature formula. We suppose that there are as many unknowns as equations. For searching solutions to this system, we construct a family of systems depending continuously on a parametert: $$Q[f_i (t)] = I[f_i (t)] i = 1,...,d,$$ coinciding with the previous system fort=1 and whose solutions att=0 are easily computed. The solution curves originating from these solutions are followed numerically and may yield a solution fort=1.     

State feedback with fractional integral control design based on the Bode’s ideal transfer function

State feedback technique through a gain matrix has been a well-known method for pole assignment of a linear system. The technique could encounter a difficulty in eliminating the steady-state errors in some states. Introducing an integral element can effectively eliminate these errors. State feedback with fractional integral control is proposed, in this work, for pole placement of a linear time invariant system. The proposed method yields simple gain formulae. The paper presents the derivation of the design formulae. The method is applied to stabilise an inherently unstable inverted pendulum-cart system. Simulation and experimental results show the effectiveness of the proposed method for set-point tracking, disturbance rejection and stabilising the inverted pendulum. Comparison with the results obtained from applying Achermann’s formula is also presented.     

Representations for vacuum potentials in stellarators

A set of vacuum potentials which has proved to be of practical use for toroidal stellarators is given by several representations: recursively by integration and by an algorithmic formula generator written in REDUCE, nonrecursively by complex loop integrals and by generating functions followed by a general explicit algebraic representation. Test examples include a collection of formulae for harmonics of low orders, a simple approximate magnetic surface invariant in special cases and a demonstration of the suppression of magnetic island by resonant fields.     

Approximate Calculation of Triple Integrals of Rapidly Oscillating Functions with the Use of Lagrange Polynomial Interflation

The authors consider the cubature formula for the approximate calculation of triple integrals of rapidly oscillating functions by using Lagrange polynomial interlineation of functions with the optimal choice of the nodal planes for the approximation of the non-oscillating set. The error of the cubature formula is estimated on the class of differentiable functions defined on a unit cube.     

Legendre expansion method for the solution of the second-and fourth-order elliptic equations

This paper presents a formula expressing Legendre polynomials in terms of their derivatives and a formula expressing a Legendre polynomial integrated k-times in terms of Legendre polynomials. In view of these formulae, the second- and fourth-order elliptic equation were solved. Moreover, the suggested method is applicable for a wide area of differential equations. The present results are in satisfactory agreement with the exact solutions.     

A New Derivation of the Cubature Kalman Filters

This paper investigates the cubature Kalman filtering (CKF) for nonlinear dynamic systems. This third‐degree rule based filter employs a spherical‐radial cubature rule to numerically compute the integrals encountered in nonlinear filtering problems, thereby removing the requirements of explicitly computing the Jacobians. The cubature rule, however, requires computing the intractable integrals over a high‐dimensional spherical region for multidimensional applications. Moreover, the cubature formula that has been used to construct the spherical cubature formula has some demerits, most notably its inconvenient properties in computation and low estimation accuracy. Aimed at these issues, a general class of CKFs that uses only cubature rules is derived in this paper. It can be shown that the conventional CKF is a special case of the proposed algorithm. The paper also includes higher‐degree CKFs, especially two representative types of the fifth‐degree CKFs. Performance of the proposed algorithms is demonstrated via two target tracking problems. The experimental results, presented herein, illustrate the superior performance of higher‐degree CKFs to conventional nonlinear filters.     

图像关于边缘提取算子的微分不变性

图像边缘是一类重要特征,该文研究图像关于边缘提取算子的微分不变性,其目的是考察图像在边缘提取的过程失去了什么信息,利用Ｌｉｅ变换群在节丛上的延拓公式以及由此导出的相关微分方程的不变性条件,文中给出了构造Ｌａｐｌａｃｅ边缘算子微分不变量的详细推导过程,为使该文自封闭,作者发展了所需的数学工具,包括一系列延拓Ｌｉｅ悍数的计算公式和有关微分方程解的不变性条件,最后给出了一个包含尺度、灰度、放置变换的数值     

A note on sampling and parameter estimation in linear stochasticsystems

Numerical differentiation formulas that yield consistent least squares parameter estimates from sampled observations of linear, time invariant higher order systems have been introduced previously by Duncan et al. (1994). The formulas given by Duncan et al. have the same limiting system of equations as in the continuous time case. The formula presented in this note can be characterized as preserving asymptotically a partial integration rule. It leads to limiting equations for the parameter estimates that are different from the continuous case, but they again imply consistency. The numerical differentiation formulas given here can be used for an arbitrary linear system, which is not the case in the previous paper by Duncan et al     

基于矩阵李群表示及容积卡尔曼滤波的视觉惯导里程计新方法

针对滤波方法实现的视觉-惯导里程计(VIO)问题,为更准确传递旋转运动的不确定性并降低系统线性化误差,提高位姿估计的精度,设计并实现了一种高维矩阵李群表示的采用容积卡尔曼滤波框架实现的VIO算法.算法将状态变量构建为一个高维李群矩阵,并定义了李群变量在容积点采样过程中的‘加法’运算,将容积点和状态均值、方差等概念由欧氏空间扩展到流形空间;采用容积变换传递状态均值及方差,避免了旋转运动复杂的雅克比矩阵计算过程,降低了模型线性化误差.最后,使用EuRoc MAV数据集进行算法验证,结果表明所提出算法在提高位姿估计精度方面是有效的.     

Consistency and convergence of general linear multistep variable stepsize variable formula methods

Linear multistep (LM) formulae are commonly used in the numerical solution of initial value problems of first order ordinary differential equations (ODE's). A rigorous theory for LM formulae, when these are implemented as constant stepsize constant formula methods, was developed after the publication of Dahlquist's classical paper [1] in 1956. After 1969 LM formulae have often been applied in practical codes as variable stepsize variable formula methods (VSVFM's). Therefore the development of a rigorous theory for LM formulae also in the case where these are used as VSVFM's is desirable. A formal definition of general LM VSVFM's is given in this paper. Then some theorems concerning the consistency and the convergence of general LM VSVFM's are formulated and proved. The results obtained in this paper can be extended for one-leg VSVFM's and for VSVFM's based on predictorcorrector schemes of different types.     

On the computation of multi-dimensional¶single layer harmonic potentials¶via approximate approximations

Multidimensional surface potentials associated with elliptic differential operators are defined by surface integrals involving fundamental solutions of the differential operators which become singular when the observation point approaches the surface. Here we combine the choice of basis functions for the so-called approximate approximation of the surface layer density with the integration of the basis functions over the tangential space by the use of appropriate asymptotic expansions. Our approach leads to cubature formulae involving only nodes of a regular grid. These formulae turn out to be extremely efficient provided the saturation error of the approximate approximation is a priori chosen sufficiently small. Received: August 2002 / Accepted: November 2002