导　　师： 朱晓临

学科专业： G0102

授予学位： 硕士

作　　者： ;

机构地区： 合肥工业大学

摘　　要： 有理函数插值理论及其应用是有理逼近研究的重要组成部分，其在唯一性、算法及误差估计等方面均取得了很多研究成果，尤其在算法的研究上更是如此。然而对于任意事先给定的插值条件，有理插值函数并不总是存在的。而其他结果诸如唯一性、算法、误差估计等，在叙述其结论时也总是假定所讨论的有理插值函数是存在的。如果存在性问题得不到很好的解决，则势必影响这些结果在使用上的确定性。规范b基即最优规范的全正基，因其具有凸包性、仿射不变性、最优保形性、端点插值性及b算法等重要性质，在cagd中起着重要的作用。cagd中广泛使用的表示曲线曲面的基函数，如：bernstein基、b样条基、nurbs基等均为规范b基。 本文对有理插值的存在性进行了研究，并给出了一类有理空间中的规范b基，在第一章回顾了有理插值的存在性和cagd中的规范b基的研究背景及研究现状。 第二章分析了有理插值出现不可达点的原因，在引入判定不可达点的定理的基础上，给出了两种解决不可达点方法，并将其推广到二元情形。 第三章给出了一元和二元的两种thiele-werner型有理插值的分块算法。并给出了具有“洞形”结构的矩形域上的应用算法。 第四章是cagd中规范b基的综述，介绍了一些规范b基理论，性质，构造和相应的b算法等。 第五章在一类有理函数空间中构造了一组规范b基，讨论了其性质和在曲线曲面造型中的应用。 Rational function interpolation theory and its application are an important part in research on rational approximation. There have been a lot of achievements in uniqueness, algorithms, error estimate and etc., especially in algorithms. But there doesn't always exist rational interpolation function for arbitrary interpolation conditions given in advance. Moreover, other results such as uniqueness, algorithms and error estimate are given which bases on that rational interpolation function exists. If the existence can't be settled well, the determinacy of these results will be influenced. Normalized B-basis, namely optimal normalized totally positive basis, plays an important role in CAGD, for it possess positive properties such as variation diminishing, convex-hull, acne invariance, tangency to the control polygon at the endpoints and B-algorithm. The widely used basis functions in CAGD, such as Bernstein, B-spline and NURBS basis, are all normalized B-basis. In this thesis, we discuss the existence of the rational interpolation and normalized B-basis in CAGD. This thesis consists of five chapters. In chapter 1, we not only retrospect the background of the research on rational interpolants and normalized B-basis, but also retrospect the study actuality of the existence of rational interpolants and normalized B-basis . In chapter 2, through analyzing the unattainable point of Thiele fractions interpolation , the method for testing the unattainable points is given. Then we give two metheds of changing the unattainable points into attainable points. The chapter 3, we give two kinds of algorithms of dividing interpolation nodes into subsets for Thiele-Werner interpolation. An efficient algorithm for computing bivariate lacunary rational interpolation is constructed. The chapter 4, we mainly discuss the properties, existence, construction and B-algorithm of normalized B-basis. The chapter 5, we give the normalized B-basis in a kind of rational space, and discuss the properties and

关 键 词： 有理函数插值 规范 基 有理插值存在性

领　　域： [理学] [理学]