两类良距分布点上Berrut有理插值的逼近性质

发布时间：2018-05-22 17:49

本文选题：重心有理插值 + Berrut有理插值　；参考：《河北师范大学》2017年硕士论文

【摘要】：重心有理插值计算量小数值稳定性好,是逼近领域研究的热点.Berrut有理插值是最常用的重心有理插值.当插值节点是良距分布点时,在这些插值节点上Berrut有理插值的Lebesgue常数关于节点个数呈对数增长.鉴于正则分布函数生成的点都是良距分布点,本文将从分布函数角度出发,分别对等距分布函数和对数分布函数这两种正则分布函数进行M(?)bius变换.通过对这两种函数中的变量进行变换和对这两种函数的整体进行变换,得到两类新的分布函数.本文主要研究变换后这两类分布函数的正则性以及由此生成的两类插值节点上Berrut有理插值的逼近性质.取得的主要研究成果如下:一方面,对函数中的变量进行M(?)bius变换,构造出具有凸性和退化性的-类分布函数,并证明其具有正则性.进而,利用对称性构造出对称-类正则分布函数.基于-类良距分布点与对称-类良距分布点,分析了这些点上Berrut有理插值的逼近性质,给出Lebesgue常数上界.最后通过数值实验比较这些良距分布点上的Lebesgue常数,实验结果显示-对数分布点、对称-等距分布点和对称-对数分布点上的插值逼近性要优于经典的等距分布点,其中对称-对数分布点上的逼近性最优.另一方面,对函数的整体进行M(?)bius变换,构造出另一类分布函数,即拟-类分布函数,该函数同样具有凸性和退化性.基于对称思想构造出对称拟-类分布函数,并证明拟-类分布函数和对称拟-类分布函数的正则性.研究了拟-类良距分布点和对称拟-类良距分布点上Berrut有理插值的逼近性,并给出Lebesgue常数上界.数值实验显示拟-类良距分布点和对称拟-类良距分布点上的逼近性质优于等距分布点,且与第二章逼近性质最优的对称-对数分点比较,当在一定范围内取值时,对称拟-对数分布点上的Berrut有理插值的逼近性质优于对称-对数分布点.
[Abstract]:Barycentric rational interpolation is the most commonly used barycentric rational interpolation because it has good numerical stability and is a hot topic in the field of approximation. When the interpolation nodes are well-spaced distribution points, the Lebesgue constant of Berrut rational interpolation on these nodes increases logarithmically with respect to the number of nodes. In view of the fact that the points generated by the regular distribution function are well-spaced distribution points, this paper will apply M(?)bius transformation to the isometric distribution function and the logarithmic distribution function respectively from the point of view of the distribution function. Through the transformation of the variables in the two functions and the global transformation of the two functions, two kinds of new distribution functions are obtained. In this paper, we study the regularity of these two kinds of distribution functions after transformation and the approximation properties of Berrut rational interpolation on two classes of interpolation nodes. The main research results are as follows: on the one hand, the variables in the function are transformed by M(?)bius, and the convexity and degeneracy of the class distribution function are constructed, and its regularity is proved. Furthermore, the symmetry-like regular distribution function is constructed by using symmetry. The approximation properties of Berrut rational interpolation on these points are analyzed, and the upper bound of Lebesgue constant is given. Finally, the Lebesgue constants on these well-spaced distribution points are compared by numerical experiments. The experimental results show that the interpolation approximation at the -logarithmic distribution points, the symmetric isometric distribution points and the symmetric -logarithmic distribution points is superior to the classical isometric distribution points. Among them, the approximation on symmetric-logarithmic distribution points is optimal. On the other hand, we construct another kind of distribution function, that is, quasi-like distribution function, by M(?)bius transformation on the whole. The function has convexity and degeneracy as well. Based on the symmetry thought, the symmetric quasi-class distribution function is constructed, and the regularity of the quasi-class distribution function and the symmetric quasi-class distribution function is proved. In this paper, we study the approximation of Berrut rational interpolation on quasi-good distance distribution and symmetric quasi-good distance distribution, and give the upper bound of Lebesgue constant. Numerical experiments show that the approximation properties of the quasi-good distance distribution point and the symmetric quasi-good distance distribution point are superior to the isometric distribution point, and compared with the symmetric logarithmic distribution point, which is the best approximation property in Chapter 2, when the value is taken within a certain range, The approximation property of Berrut rational interpolation on symmetric quasi-logarithmic distribution points is superior to that of symmetric logarithmic distribution points.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.3

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