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三次样条函数的误差阶 总被引:2,自引:0,他引:2
1.介绍 关于样条函数的误差估计是近几年来在样条函数理论中研究较多的课题之一,而关于误差的阶,是被人们特别注意的对象。在三次样条函数的研究中,关于零阶、一阶,二阶导数的误差已有较好的结果,关于三阶导数的误差虽亦有种种估计,但是,从阶讲还未达到令人满意的结果。J.L.Walsh等人曾期望能得到不依赖于步长比的估计,我们指出要使当最长的小区间长度|π|趋于零时,三阶导数的误差|e~(3)|趋于零,步长比L和 相似文献
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三次样条函数的误差估计 总被引:2,自引:0,他引:2
§1.引言 三次样条函数的误差估计十分重要,国内外已作了大量的工作.至今最好的结果是 定理1.设f(x)∈)C~m(Ω)(m=1,2,3,4),s(x)∈S~2(Ω,π)是f(x)的关于分划π的Ⅰ型三次样条函数,则 相似文献
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C.Davis和W.J.Kammerer曾先后用不同的方法证明了如下定理: 设y_0,y_1,…,y_n为实数,满足y_0>y_1,y_1y_3,…,则存在唯一的一个n次多项式P_n(x)和一组点x_0,x_1,…,x_n使得P_n(x_i)=y_i(i=0,1,…,n),P′_n(x_i)=0(i=1,2,…,n-1),0=x_0相似文献
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在理论和实践中,人们对保形拟合问题特别有兴趣、[1]与[2]讨论了二次样条的保形问题,[3]则给出有关三次样条保凸问题的一系列充分条件。 给定点列{(x_i),y_i)}(i=0,1,…,N)(也称型值).通过此点列的三次样条函数在 x=x_j处的一阶导数m_j满足方程组: 相似文献
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陈天平 《数学年刊A辑(中文版)》1983,(3)
本文通过插值样条函数研究函数的构造性质。设f(x)∈L_p[0,1],S_n(f;x)为其插值样条函数。文中给出用‖S_n~((r))(f;x)‖_p来描述f~((r))(x)∈L_p[0,1]的充要条件。对于相当一类的插值样条和缺插值样条文中的结果均能适用。 相似文献
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关于三次样条函数的两点注记 总被引:4,自引:0,他引:4
常庚哲 《数学的实践与认识》1979,(2)
关于样条函数的理论和应用,近年来,在国内一些数学刊物上已有详细介绍.本文主要做了两件事:1.采用 Hermite 插值基函数推出三次样条的两种节点关系式;2.讨论了端点条件对于样条函数的影响,特别地,改进了[3]文的结果. 相似文献
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常庚哲 《数学的实践与认识》1984,(4)
一、引言 三次样条函数的连续性条件可以通过节点上的一阶导数 m_i 或者二阶导数 M_i 所适合的线性方程组表示出来,它们分别被称为 m-关系式及 M-关系式.这两种关系式通常由不同的基函数推导出来,因此需要做两次独立的计算,见文献[1],[2]及[3].本文指出,应用 Bernstein 基函数及有关的导数公式,可经一次计算同时得出这两种关系式.本文不假定读者具有任何关于样条函数的知识. 相似文献
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Sigrid Fredenhagen Hans Joachim Oberle Gerhard Opfer 《Journal of Approximation Theory》1999,96(2):182
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. 相似文献
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A cubic spline approximation to the heat conduction equationis shown to correspond to a special case of a finite-differencescheme considered by Saul'yev. The spline approximation producesat each time level a spline function which may be used to obtainthe solution at any point in the range of the space variable. 相似文献
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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]. 相似文献
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Dedicated to Professor M. J. D. Powell on the occasion
of his sixty-fifth birthday and his retirement.
In this paper, we design differentiable, two-dimensional, piecewise polynomial cubic prewavelets of particularly small compact
support. They are given in closed form, and provide stable, orthogonal decompositions of L
2
(R
2
) . In particular, the splines we use in our prewavelet constructions give rise to stable bases of spline spaces that contain
all cubic polynomials, whereas the more familiar box spline constructions cannot reproduce all cubic polynomials, unless resorting
to a box spline of higher polynomial degree. 相似文献
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The use of polynomial splines as a basis for the interpolationof discrete data can be theoretically justified by a minimumprinciple. It is natural to apply this principle also if shapepreserving is added as a constraint, although the constructionprocess is then nonlinear. We discuss two algorithms for theconstruction of the cubic spline interpolant under the constraintof positivity or monotonicity, and give a detailed convergenceanalysis. Numerical tests illustrate that analysis. 相似文献