具有承袭性的高阶导数有理插值算法

荆科; 刘业政; 康宁

doi:10.3879/j.issn.1000-0887.2014.08.009

具有承袭性的高阶导数有理插值算法

doi: 10.3879/j.issn.1000-0887.2014.08.009

荆科^1,2,
刘业政¹,
康宁³

¹合肥工业大学管理学院, 合肥 230009;²阜阳师范学院数学与金融学院, 安徽阜阳 236037;³阜阳师范学院经济与管理学院, 安徽阜阳 236037

基金项目: 国家重点基础研究发展计划（973计划）（2013CB329603）

详细信息

作者简介:
荆科（1983—），男，安徽颍上人，博士生（E-mail: jingxuefei296@sina.com）；刘业政（1965—），男，安徽和县人，教授，博士生导师（通讯作者. E-mail: liuyezheng@hfut.edu.cn）.

中图分类号: O241.3;O174.42
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- 被引次数: 0
出版历程
- 收稿日期: 2013-12-06
- 修回日期: 2014-06-09
- 刊出日期: 2014-08-15

High Order Derivative Rational Interpolation Algorithm With Heredity

¹School of Management, Hefei University of Technology, Hefei 230009, P.R.China;²School of Mathematics and Finance, Fuyang Teachers College, Fuyang, Anhui 236037, P.R.China;³School of Economics and Management, Fuyang Teachers College, Fuyang, Anhui 236037, P.R.China

Funds: The National Basic Research Program of China (973 Program)（2013CB329603）

摘要

摘要: 切触有理插值是函数逼近的一个重要内容，而降低切触有理插值的次数和解决切触有理插值函数的存在性是有理插值的一个重要问题.切触有理插值函数的算法大都是基于连分式进行的，其算法可行性是有条件的，且计算量较大.利用Newton(牛顿)多项式插值的承袭性和分段组合的方法，构造出了一种无极点且满足高阶导数插值条件的切触有理插值函数，并推广到向量值切触有理插值情形；既解决了切触有理插值函数存在性问题，又降低了切触有理插值函数的次数.最后给出误差估计，并通过数值实例说明该算法具有承袭性、计算量低、便于编程等特点.
- Lauwerier映射 /
- 反演极限空间 /
- 上半连续分解 /
- Markov分割 /
- 拓扑半共轭
Abstract: Osculatory rational interpolation was an important theme of function approximation, meanwhile, reducing the degree and solving the existence of the osculatory rational interpolation function made a crucial problem for rational interpolation. The previous algorithms of osculatory rational interpolation functions mostly depended on the continued fraction with conditional feasibility and high computation complexity. Based on heredity of the Newton interpolation and the method of piecewise combination, an osculatory rational interpolation function without real poles was constructed to meet the condition of high order derivative interpolation, and was in turn extended to the vector-valued cases. It not only solved the existence problem for the osculatory rational interpolation function, but reduced the degree of the rational function. Furthermore, the error estimates of the new algorithm was given. Results of the numerical examples illustrate the new algorithm’s heredity, low computation complexity and easy programmability.
- osculatory rational interpolation /
- Newton interpolation /
- piecewise combination /
- heredity /
- high order derivative

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参考文献(14)

[1]	Salzer H E. Note on osculatory rational interpolation[J].Mathematics of Computation,1962,16(80): 486-491.
[2]	Wuytack L. On the osculatory rational interpolation problem[J].Mathematics of Computation,1975,29(131): 837-843.
[3]	朱晓临. (向量)有理函数插值的研究及其应用[D]. 博士学位论文. 合肥: 中国科学技术大学, 2002: 32-43.(ZHU Xiao-lin. Research on (vector) rational function interpolation and its application[D]. PhD Thesis. Hefei: University of Science and Technology of China, 2002: 32-43.(in Chinese))
[4]	王仁宏, 朱功勤. 有理函数逼近及其应用[M]. 北京: 科学出版社, 2004: 117-183.(WANG Ren-hong, ZHU Gong-qin.Rational Function Approximation and Its Application [M]. Beijing: Science Press, 2004: 117-183.(in Chinese))
[5]	朱功勤, 马锦锦. 构造切触有理插值的一种方法[J]. 合肥工业大学学报(自然科学版), 2006,29(10): 1320-1326.(ZHU Gong-qin, MA Jin-jin. A way of constructing osculatory rational interpolation[J].Journal of Hefei University of Technology(Natural Sciences),2006,29(10): 1320-1326.(in Chinese))
[6]	朱功勤, 何天晓. 具有重节点的分段Padé逼近的一个算法[J]. 计算数学, 1981,3(2): 179-182.（ZHU Gong-qin, HE Tian-xiao. A method of calculation about theN -point sectional Padé approximant[J].Mathematica Numerica Sinica,1981,3(2): 179-182.(in Chinese))
[7]	朱功勤, 黄有群. 插值(切触)分式表的构造[J]. 计算数学, 1983,5(3): 310-317. (ZHU Gong-qin, HUANG You-qun. The construction of the table of interpolating (osculatory) rationals[J].Mathematica Numerica Sinica,1983,5(3): 310-317.(in Chinese))
[8]	苏家铎, 黄有度. 切触有理插值的一个新算法[J]. 高等学校计算数学学报, 1987,9(2): 170-176.(SU Jia-duo, HUANG You-du. A new algorithm of osculatory rational interpolation[J].Numerical Mathematics: A Journal of Chinese Universities,1987,9(2): 170-176.(in Chinese))
[9]	荆科, 康宁, 姚云飞. 一种切触有理插值的构造方法[J]. 中国科学技术大学学报, 2013,43(6): 477-479.(JING Ke, KANG Ning, YAO Yun-fei. A new method of constructing osculatory rational interpolation function [J].Journal of University of Science and Technology of China,2013,43(6): 477-479.(in Chinese))
[10]	朱功勤, 顾传青. 向量的Salzer定理[J]. 数学研究与评论, 1990,10(4): 516.(ZHU Gong-qin, GU Chuan-qing. Vector Salzer theorem[J].Journal of Mathematical Research and Exposition,1990,10(4): 516.(in Chinese))
[11]	陶有田, 朱晓临, 周金明, 徐鑫. 向量值切触有理插值存在性的一种判别方法[J]. 合肥工业大学学报（自然科学版）, 2007,30(1): 117-120.(TAO You-tian, ZHU Xiao-lin, ZHOU Jin-ming, XU Xin. A decision method for existence of vector-valued osculatory rational interpolants[J].Journal of Hefei University of Technology (Natural Sciences),2007,30(1): 117-120.(in Chinese))
[12]	Sidi A. A new approach to vector-valued rational interpolation[J].Journal of Approximation Theory,2004,130(2): 179-189.
[13]	Sidi A. Algebraic properties of some new vector-valued rational interpolants[J].Journal of Approximation Theory,2006,141(2): 142-161.
[14]	盛中平, 林正华. 广义 Vandermonde 行列式及其应用[J]. 高等学校计算数学学报, 1996,18(3): 217-225.(SHENG Zhong-ping, LIN Zheng-hua.The generalized Vandermonde determinant and its applications [J].Numerical Mathematics: A Journal of Chinese Universitie s, 1996,18(3): 217-225.(in Chinese))