Sextic C 1-spline collocation methods for solving delay differential equations

In this paper, we propose a global collocation method for the numerical solution of the delay differential equations (DDEs). The method presented is based on sextic C ¹-splines s(x) as an approximation to the exact solution y(x) of the DDEs. Convergence results shows that the error bounds ‖ s ^j ?y ^j ‖=O(h ⁶), j=0, 1, in the uniform norm. Moreover, the stability analysis properties of these methods have been studied. Numerical experiments will also be considered.     

Spline approximation with interpolation constraints for numerical integration

By using a Walsh-type theorem, we impose a finite number of interpolation constraints to a polynomial spline approximation operator. These constrained approximating splines reproduce polynomials and give orders of convergence identical to those of the unconstrained ones. We give an application in numerical integration by using the constrained splines for evaluation of Cauchy principal value integrals.     

Fair, G- and C-continuous circle splines for the interpolation of sparse data points

This paper presents a carefully chosen curve blending scheme between circles, which is based on angles, rather than point positions. The result is a class of circle splines that robustly produce fair-looking G²-continuous curves without any cusps or kinks, even through rather challenging, sparse sets of interpolation points. With a simple reparameterization the curves can also be made C²-continuous. The same method is usable in the plane, on the sphere, and in 3D space.     

Exponentially fitted cubic spline for two-parameter singularly perturbed boundary value problems

In this paper, we derive an exponentially fitted difference scheme using cubic splines for a singularly perturbed ordinary differential equation with two small parameters affecting the convection and diffusion terms. The solution of the problem exhibits a boundary layer on the left-hand side of the domain. Bounds on the derivatives of the solution are derived. A first-order numerical method is constructed. Numerical results are presented to establish the theoretical results and to show the efficiency of the method.     

有理[2m+1,2m]型分段插值样条

常见的较低次有理带单形状因子分段有理插值样条通过代数运算,可用Bernstein基函数等价表示,这类分段插值样条利用Hermite插值的方法推广到高次有理[2m+1,2m]型,样条的生成曲线满足Cm-连续,并给出了具体的Bern-stein基函数表示方法的表达式,其形式较为简单,最后分别讨论了这类有理插值的逼近阶与约束域及保单调等方面的形状因子的选取情况,并给出了例子分析。     

A class of C 2piecewise quintic polynomials

A new class of C ² piecewise quintic interpolatory polynomials is defined. It is shown that this new class contains a number of interpolatory functions which present practical advantages, when compared with the conventional cubic spline.     

Reevaluation of daylight spectral distributions

The spectral power distribution of colorimetric daylight illuminants was determined based on measurements1 done some 40 years ago. The data were averaged and standardized at every full 10 nm. Later, to follow colorimetric practice, these data have been interpolated (linearly) to every 5 nm, but the factors to calculate the values have not been changed. We show in this article that this is not correct and leads to some discrepancies, especially if data with even smaller step size are required. We suggest that when correcting this discrepancy one should at the same time go from the linear interpolation technique, a rather crude method for a spectral distribution with many minor minima and maxima in the spectrum, to a nonlinear one. We show that a third‐order spline interpolation can provide smooth functions resembling the original data to a high degree. © 2000 John Wiley & Sons, Inc. Col Res Appl, 25: 250–259, 2000     

三次样条插值在称重仪表误差补偿中的应用

介绍了采用三次样条曲线插值方法对称重仪表的非线性误差进行修正的方法,该修正方法对于线性误差较大的传感器有很好的补偿效果,在某些测量精度较高的场合采用三次样条曲线插值可以获得比流行的多段折线线性补偿方法更高的精度。通过对一款5位半仪表采用三种误差处理方法比较,验证了三次样条曲线插值误差补偿方法对传感器误差的补偿效果。该方法对于其他的数据采集系统精度提升有一定的借鉴意义。     

一类具有线性分母的有理插值样条的逼近问题

对被插函数f(t)∈c^3[t0,tm]，讨论了一类具有线性分母的有理插值函数的逼近性质，证明了插值误差系数的有界性和其对参数的双重对称性。     

Analyzing angle crashes at unsignalized intersections using machine learning techniques

A recently developed machine learning technique, multivariate adaptive regression splines (MARS), is introduced in this study to predict vehicles’ angle crashes. MARS has a promising prediction power, and does not suffer from interpretation complexity. Negative Binomial (NB) and MARS models were fitted and compared using extensive data collected on unsignalized intersections in Florida. Two models were estimated for angle crash frequency at 3- and 4-legged unsignalized intersections. Treating crash frequency as a continuous response variable for fitting a MARS model was also examined by considering the natural logarithm of the crash frequency. Finally, combining MARS with another machine learning technique (random forest) was explored and discussed. The fitted NB angle crash models showed several significant factors that contribute to angle crash occurrence at unsignalized intersections such as, traffic volume on the major road, the upstream distance to the nearest signalized intersection, the distance between successive unsignalized intersections, median type on the major approach, percentage of trucks on the major approach, size of the intersection and the geographic location within the state. Based on the mean square prediction error (MSPE) assessment criterion, MARS outperformed the corresponding NB models. Also, using MARS for predicting continuous response variables yielded more favorable results than predicting discrete response variables. The generated MARS models showed the most promising results after screening the covariates using random forest. Based on the results of this study, MARS is recommended as an efficient technique for predicting crashes at unsignalized intersections (angle crashes in this study).