OPTIMAL ERROR BOUNDS FOR THE CUBIC SPLINE INTERPOLATION OF LOWER SMOOTH FUNCTIONS(1)

In this paper,the kernel of the cubic spline interpolation is given.An optimal error bound for the cu-bic spline interpolation of lower smooth functions is obtained.     

样条型矩阵有理插值

The matrix valued rational interpolation is very useful in the partial realization problem and model reduction for all the linear system theory. Lagrange basic functions have been used in matrix valued rational interpolation. In this paper, according to the property of cardinal spline interpolation, we constructed a kind of spline type matrix valued rational interpolation, which based on cardinal spline. This spline type interpolation can avoid instability of high order polynomial interpolation and we obtained a useful formula.     

On cubic Hermite coalescence hidden variable fractal interpolation functions

Hermite interpolation is a very important tool in approximation theory and numerical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set,and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the shortcoming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a C1-cubic Hermite interpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global C2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an alternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15(1)(2007), pp. 41-53].     

一类三角域上的三次样条插值(英文)

In this paper, we consider spaces of cubic C^1-spline on a class of triangulations. By using the inductive algorithm, the posed Lagrange interpolation sets are constructed for cubic spline space. It is shown that the class of triangulations considered in this paper are nonsingular for S1/3 spaces. Moreover, the dimensions of those spaces exactly equal to L. L. Schuraaker＇s low bounds of the dimensions. At the end of this paper, we present an approach to construct triangulations from any scattered planar points, which ensures that the obtained triangulations for S1/3 space are nonsingular.     

Deficient cubic splines with average slope matching

We obtain a deficient cubic spline function which matches the functions with certain area matching over a greater mesh intervals, and also provides a greater flexibility in replacing area matching as interpolation. We also study their convergence properties to the interpolating functions.     

ERROR BOUNDS IN PERIODIC QUARTIC SPLINE INTERPOLATION

In this paper we develop periodic quartic spline interpolation theory which,in general,gives better fus to continuous functions than does the existing quintic spline interpolation theory.The main theorem of the paper is to establish that r=0,1,2,3.Also,the nanperiodic cases cannot be constructed empoly-ing the methodology of this paper because that will involve several other end conditions entirely different than(1,10).     

W_2~m空间中样条插值算子与线性泛函的最佳逼近

In this paper,the convergency of spline interpolation operators is obtained,these spline operators are determined by linear differential operators and constraint functionals.The errors of the interpolating spline with EHB fanctionals are estimated.The best approximation of linear functionals on W2^n spaces are investigated,which let to a useful computational method for the approximation solution of higher order linear differential equations with multipoint boundary value conditions.     

BASES OF BIVARIATE SPLINE SPACES WITH CROSS-CUT GRID PARTITIONS

In this paper we determine the dimensions of certain bivariate spline spaces with cross-cut grid partitions and give explicit expressions of their bases functions. As an immediate consequence, the closures of these spaces over all grid partitions of the same type are known. The results in this paper can be applied to interpolation and approximation by bivariate spline functions.     

On the Rational Spline Functions

Since Schoenberg's fundamental paper was published in 1946, the theory of polynomial spline functions developed rapldly and was rich in contents. Spline functions have wide range of applications to numerical approximation.But to certain functions, for example, to the function with singular points, the interpolation by polynomial spline is not always efficient and reasonable. Since 1973 some authors have investigated the rational spline functions, and abtained a lot of results. But     

UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS

In this paper, we are concerned with uniform superconvergence of Galerkin methods for singularly perturbed reaction-diffusion problems by using two Shishkin-type meshes. Based on an estimate of the error between spline interpolation of the exact solution and its numerical approximation, an interpolation post-processing technique is applied to the original numerical solution. This results in approximation exhibit superconvergence which is uniform in the weighted energy norm. Numerical examples are presented to demonstrate the effectiveness of the interpolation post-processing technique and to verify the theoretical results obtained in this paper.     

Comparative Analysis of Cubic Spline and Kernel Estimation of a Probit Function

The least-squares cubic spline and the kernel estimators produce comparable mean squared errors, although the kernel produces smaller mean squared errors when the variable increases away from 0. Mean squared error increases with an increase in the number of knots (for the cubic spline) or reduced band width (for the kernel estimator). The cubic spline produces smaller mean squared errors when all observations are made at knots than when they are spaced out between knots. Irrespective of the exact form of the probit function g(x), the cubic spline estimator is asymptotically unbiased, while the kernel estimator only converges to g(x) under certain conditions. Moreover, the cubic spline is a smooth function, which is twice differentiable on the interval [0,1].     

CAN A CUBIC SPLINE CURVE BE G3

This paper proposes a method to construct an G³cubic spline curve from any given open control polygon.For any two inner Bezier points on each edge of a control polygon,we can de ne each Bezier junction point such that the spline curve is G2-continuous.Then by suitably choosing the inner Bezier points,we can construct a global G³spline curve.The curvature combs and curvature plots show the advantage of the G³cubic spline curve in contrast with the traditional C2 cubic spline curve.     

几种有理插值函数的逼近性质

1　引　　言在曲线和曲面设计中,样条插值是有用的和强有力的工具.不少作者已经研究了很多种类型的样条插值[1,2,3,4].近些年来,有理插值样条,特别是三次有理插值样条,以及它们在外型控制中的应用,已有了不少工作[5,6,7].有理插值样条的表达式中有某些参数,正是由于这些参数,有理插值样条在外型控制中充分显示了它的灵活性;但也正是由于这些参数,使它的逼近性质的研究增加了困难.因此,关于有理插值样条的逼近性质的研究很少见诸文献.本文在第二节首先叙述几种典型的有理插值样条,其中包括分母为一次、二次的三次有理插值样条和仅基于函数值…     

C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves, the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed. The quartic Hermite spline curves not only have the same interpolation and conti-nuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C2 continuity by the shape parameters when the interpolation conditions are fixed.     

Optimal error bounds for the cubic spline interpolation of lower smooth functions (1)

In this paper, the kernel of the cubic spline interpolation is given. An optimal error bound for the cubic spline interpolation of lower smooth functions is obtained.     

Discrete cubic spline interpolation

Defining equations, a best approximation property, and error bounds are given for a discrete cubic spline interpolant. Furthermore the distance between two cubic spline interpolants is estimated, and numerical examples are provided.     

可调形三次三角Cardinal插值样条曲线

在三次Cardinal插值样条曲线的基础上,引入了三角函数多项式,得到一组带调形参数的三次三角Cardinal样条基函数,以此构造一种可调形的三次三角Cardinal插值样条曲线.该插值样条可以精确表示直线、圆弧、椭圆以及自由曲线,改变调形参数可以调控插值曲线的形状.该插值样条避免了使用有理形式,其表达式较为简洁,计算量也相对较少,从而为多种线段的构造与处理提供了一种通用与简便的方法.     

Galerkin's finite element formulation of the system of fourth‐order boundary‐value problems

In this article, a Galerkin's finite element approach based on weighted‐residual is presented to find approximate solutions of a system of fourth‐order boundary‐value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece‐wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non‐polynomial spline and cubic non‐polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1551–1560, 2011     

C~3连续的三角样条函数与曲线

本文构造了一种三次三角样条函数 ,函数的每一段由三个函数值生成 ,具有C3连续性和较好的逼近性 ,可方便地进行插值 .基于同样的方法得出了一种C3连续的三角样条曲线 ,曲线也有较好的逼近性 ,而且具有局部性、保凸性等特性 .     

基于三次样条插值函数的非线性动力系统数值求解

三次样条插值函数具有良好的收敛性、稳定性与二阶光滑性.研究了借助三次样条插值函数构造的非线性动力系统数值求解方法,分析了该方法与已有的非线性动力系统数值求解方法的优缺点,刻画了误差估计且给出了数值算例.结果表明基于三次样条插值函数构造的数值方法比已有的方法收敛速度快、逼近精度高且能够很好地逼近非线性动力系统的解析解.