Cubic spline interpolation with quasiminimal B-spline coefficients

Summary The end conditions for cubic spline interpolation with equidistant knots will be defined so as to make the (slightly modified) B-spline coefficients minimal. This produces good approximation results as compared e.g. with the not-a-knot spline.     

On the Construction of Optimal Monotone Cubic Spline Interpolations

In this paper we derive necessary optimality conditions for an interpolating spline function which minimizes the Holladay approximation of the energy functional and which stays monotone if the given interpolation data are monotone. To this end optimal control theory for state-restricted optimal control problems is applied. The necessary conditions yield a complete characterization of the optimal spline. In the case of two or three interpolation knots, which we call thelocalcase, the optimality conditions are treated analytically. They reduce to polynomial equations which can very easily be solved numerically. These results are used for the construction of a numerical algorithm for the optimal monotone spline in the general (global) case via Newton's method. Here, the local optimal spline serves as a favourable initial estimation for the additional grid points of the optimal spline. Some numerical examples are presented which are constructed by FORTRAN and MATLAB programs.     

End conditions for improved cubic spline derivative approximations

We consider the problem of deriving accurate end conditions for cubic spline interpolation at equally spaced knots. In particular we derive a number of end conditions which lead to derivative approximations of high accuracy.     

关于三次插值样条函数的局部性质

近年来,三次插值样条函数的局部性质,受到了人们的注意,翁祖荫考察了在一般边界条件下三次插值样条的局部逼近性质及插值样条导数的局部界值,不过他对于分     

End conditions for interpolatory cubic splines with unequally spaced knots

A class of end conditions is derived for cubic spline interpolation at unequally spaced knots. These conditions are in terms of function values at the knots and lead to 0 (h⁴) convergence uniformly on the interval of interpolation.     

Improved sampling and reconstruction in spline subspaces

As a special shift-invariant spaces, spline subspaces yield many advantages so that there are many practical applications for signal or image processing. In this paper, we pay attention to the sampling and reconstruction problem in spline subspaces. We improve lower bound of sampling set conditions in spline subspaces. Based on the improved explicit lower bound, a improved explicit convergence ratio of reconstruction algorithm is obtained. The improved convergence ratio occupies faster convergence rate than old one. At the end, some numerical examples are shown to validate our results.     

Error bounds in periodic quartic spline interpolation

In this paper we develop periodic quartic spline inter polation theory which, in general, gives better fits to continuous functions than does the existing quintic spline inter polation theory. The main theorem of the paper is to establish that ⋎s^(r)-y^(r)⋎=O(h^6−r), r=0,1,2,3. Also, the nonperiodic cases cannot be constructed empolying the methodology of this paper because that will involve several other end conditions entirely different than (1.10).     

OnG 2 continuous cubic spline interpolation

In this paper the interpolation byG ² continuous planar cubic Bézier spline curves is studied. The interpolation is based upon the underlying curve points and the end tangent directions only, and could be viewed as an extension of the cubic spline interpolation to the curve case. Two boundary, and two interior points are interpolated per each spline section. It is shown that under certain conditions the interpolation problem is asymptotically solvable, and for a smooth curvef the optimal approximation order is achieved. The practical experiments demonstrate the interpolation to be very satisfactory. Supported in prat by the Ministry of Science and Technology of Slovenjia, and in part by the NSF and SF of National Educational Committee of China.     

An algorithm for the interpolation of functions using quintic splines

A method is described for the interpolation of N arbitrarily given data points using fifth degree polynomial spline functions. The interpolating spline is built from a set of basis functions belonging to the fifth degree smooth Hermite space. The resulting algebraic system is symmetric and bloc-tridiagonal. Its solution is calculated using a direct inversion method, namely a block-gaussian elimination without pivoting. Various boundary conditions are provided for independently at each end point. The stability of the algorithm is examined and some examples are given of experimental convergence rates for the interpolation of elementary analytical functions. A listing is given of the two FORTRAN subroutines INSPL5 and SPLIN5 which form the algorithm.     

Spline prewavelets for non-uniform knots

Summary In this paper, we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non-uniform knot-sequences. To this end, we show, among other things, the existence of compactly supported prewavelets and of prewavelets that are globally supported, but decay exponentially. In each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets.     

基于函数值的有理三次插值样条曲线的区域控制

将插值曲线约束于给定的区域之内是曲线形状控制中的重要问题.构造了一种基于函数值的分母为三次的C~1连续有理三次插值样条.这种有理三次插值样条中含有二个调节参数,因而给约束控制带来了方便.对该种插值曲线的区域控制问题进行了研究,给出了将其约束于给定的折线、二次曲线之上、之下或之间的充分条件.最后给出了数值例子.     

Convergence analysis of nonic-spline solutions for special nonlinear sixth-order boundary value problems

In this paper, we use nonic-spline polynomial method for the numerical solution of special nonlinear sixth-order two-point boundary value problems. The main idea is to use the conditions of continuity as discretization equations for the sixth-order boundary value problem. The end conditions are derived for defined spline. A new approach for convergence analysis of the presented method discussed. Some examples are solved to illustrate the applications of method, and to compare the computed results with other existing known methods.     

A shape preserving area true approximation of histogram by rational splines

For a given histogram, we consider an application of a simple rational spline to a shape preserving area true approximation of the histogram. An algorithm for determination of the spline is as easy as one with a quadratic polynomial spline, while the latter does not always preserve the shape of the histogram. Some numerical examples are given at the end of the paper.     

Box－样条曲面的正性条件

研究了Ｂｏｘ－样条曲面的控制点与Ｂｏｘ－样条曲面的正性之间的关系．给出了Ｂｏｘ－样条曲面正性的必要条件、充分条件．由此我们得到了Ｂｏｘ－样条曲面的单调性条件，推广了Ｗ．ＤＡＨＭＥＮ和Ｃ．Ａ．ＭＩＣＣＨＥＬＬＩ在文［２］中给出的Ｂｏｘ－样条曲面的单调性结论     

整体最优双圆弧拟合

提出了用整体最优双圆弧样条拟合离散数据点的算法．首先分析了双圆弧逼近的误差分布,并且根据这个误差分布调整端点、切向和双圆弧插值时的惟一自由变量,使得数据点的误差分布均匀、圆弧的段数尽可能少,由此得到G^1连续的整体最优双圆弧样条．这个方法在数值控制刀具的运动路线的设计和机器人的移动路线设计上非常有用。     

Interpolation by splines satisfying mixed boundary conditions

We consider interpolation of Hermite data by splines of degreen withk given knots, satisfying boundary conditions which may involve derivatives at both end points (e.g., a periodicity condition). It is shown that, for a certain class of boundary conditions, a necessary and sufficient condition for the existence of a unique solution is that the data points and knots interlace properly and that there does not exist a polynomial solution of degreen?k. The method of proof is to show that any spline interpolating zero data vanishes identically, rather than the usual determinantal approach.     

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].     

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.     

两条Bézier样条的光滑连接

研究了用一条样条曲线把两条不相连接的样条曲线光滑连接起来的问题,给出了连接两条一元n次参数样条曲线为一条新的一元n次参数样条曲线的条件,适用于参数样条曲线添加控制顶点的情形,进一步得到了两条一次、二次、三次Bézier样条曲线在几何连续性下实现自然光滑连接的条件.