有理插值的Neville算法和凸组合方法构造切触有理插值
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摘要
随着多项式插值理论的日趋完善，人们发现多项式插值并不能总是很好的解决问题。对于一类有极点的函数来说，用多项式逼近的效果并不好。而采用有理分式函数来逼近，效果就好得多了。
本文第一章从一般的有理插值问题的提法入手，介绍了有理插值的基本概念和一些性质。第二章介绍了一种计算一般的有理函数插值的Neville算法。以往的文章在介绍有理插值算法时，往往更多的提及用连分式来构造有理分式函数，而较少详细介绍有理插值的Neville算法(有的文献也将之称为Stoer方法)，大多只给出大概的思想。本文将具体的介绍这种算法，给出详细的推导过程。它采用一种递推的思想，在只须计算某点的有理逼近值而无需求逼近函数的情况下，这种方法尤其高效。同样，利用这种方法，能根据需要构造出不同的有理分式函数，并且每个形式的有理分式函数都是唯一存在的。
本文第三章转入切触有理插值。所谓切触有理插值即要求插值函数是有理分式函数，满足在插值点的函数值及导数值均相等。传统的解决切触有理插值的问题依然是跟连分式相关，这种方法的缺点是计算量较大，而且要求插值点是适定的，因此可行性是有条件的。而本文将介绍的是一种凸组合的方法构造的切触有理插值，它格式简单，对插值条件没有要求，便于在计算机上实现。同时，本文还将这种方法推广到二元的情况，并给出数值例子。
本文所做的主要工作是详细给出Neville算法的推导过程，并将凸组合方法构造的切触有理插值从一元推广到二元。
So far, the theory for polynomial interpolation is nearly perfect. While we benefit from this algorithm, we still find that in certain circumstances, polynomial interpolation is not such a good choice. Instead, we can use a rational interpolation. For example, for approximation of a kind of function with one or more extremal points, rational interpolation can get better results than ordinary polynomial interpolation.
In the first chapter, we give some basic concepts and theorems of the rational interpolation for better understanding of this paper. And then in the second chapter, we introduce a quasi- Neville algorithm for solving ordinary rational interpolation. We know that most articles which introduce algorithms for solving rational interpolation adopt a continued fraction approximation method, but less mention the quasi-Neville method. So in this paper, we will give a very specific deduction for this method, which is an iterated interpolation method. Comparing to the continued fraction approximation method, this quasi-Neville method is much more effective when the expression of the approximating function is unnecessary.
In the third chapter of this paper, we introduce a convex combination method for solving osculatory rational interpolation problem. Here the definition for "osculatory rational interpolation" means that the approximating rational function should satisfy both values and derivatives at given points. The usual methods for solving this osculatory rational interpolation are also related to continued fraction approximation method, which needs both suitable points and a large amount of calculation. The advantages for the convex combination method are less calculation, no limits for points and convenience for realization by computer.
The main work of this paper is giving a specific deduction for the quasi-Neville method and generalizing a convex combination method for solving two-variable osculatory rational interpolation problem.
本文第一章从一般的有理插值问题的提法入手，介绍了有理插值的基本概念和一些性质。第二章介绍了一种计算一般的有理函数插值的Neville算法。以往的文章在介绍有理插值算法时，往往更多的提及用连分式来构造有理分式函数，而较少详细介绍有理插值的Neville算法(有的文献也将之称为Stoer方法)，大多只给出大概的思想。本文将具体的介绍这种算法，给出详细的推导过程。它采用一种递推的思想，在只须计算某点的有理逼近值而无需求逼近函数的情况下，这种方法尤其高效。同样，利用这种方法，能根据需要构造出不同的有理分式函数，并且每个形式的有理分式函数都是唯一存在的。
本文第三章转入切触有理插值。所谓切触有理插值即要求插值函数是有理分式函数，满足在插值点的函数值及导数值均相等。传统的解决切触有理插值的问题依然是跟连分式相关，这种方法的缺点是计算量较大，而且要求插值点是适定的，因此可行性是有条件的。而本文将介绍的是一种凸组合的方法构造的切触有理插值，它格式简单，对插值条件没有要求，便于在计算机上实现。同时，本文还将这种方法推广到二元的情况，并给出数值例子。
本文所做的主要工作是详细给出Neville算法的推导过程，并将凸组合方法构造的切触有理插值从一元推广到二元。
So far, the theory for polynomial interpolation is nearly perfect. While we benefit from this algorithm, we still find that in certain circumstances, polynomial interpolation is not such a good choice. Instead, we can use a rational interpolation. For example, for approximation of a kind of function with one or more extremal points, rational interpolation can get better results than ordinary polynomial interpolation.
In the first chapter, we give some basic concepts and theorems of the rational interpolation for better understanding of this paper. And then in the second chapter, we introduce a quasi- Neville algorithm for solving ordinary rational interpolation. We know that most articles which introduce algorithms for solving rational interpolation adopt a continued fraction approximation method, but less mention the quasi-Neville method. So in this paper, we will give a very specific deduction for this method, which is an iterated interpolation method. Comparing to the continued fraction approximation method, this quasi-Neville method is much more effective when the expression of the approximating function is unnecessary.
In the third chapter of this paper, we introduce a convex combination method for solving osculatory rational interpolation problem. Here the definition for "osculatory rational interpolation" means that the approximating rational function should satisfy both values and derivatives at given points. The usual methods for solving this osculatory rational interpolation are also related to continued fraction approximation method, which needs both suitable points and a large amount of calculation. The advantages for the convex combination method are less calculation, no limits for points and convenience for realization by computer.
The main work of this paper is giving a specific deduction for the quasi-Neville method and generalizing a convex combination method for solving two-variable osculatory rational interpolation problem.
引文
[1] A. Cuyt and B. Verdonk. A review of branched continued fraction theory for the construction of multivariate rational approximants. Appl.Number.Math, 19(4):263-271, 1988.
[2] J．Stoer，R．Bulirsch，孙文瑜等译．数值分析引论．南京：南京大学出版社，1995．
[3] H. E. Salze. Note on osculatory rational imerpolation. Math. Computer, 19: 486-491, 1962.
[4] J. Stoer. Uber zwei algorithmen zur interpolation mit rationalen funktionen. Numerische Mathematik, 3: 285-304, 1961.
[5] 朱功勤，马锦锦．构造切触有理插值的一种方法．合肥工业大学学报，29，2006．
[6] 王仁宏，梁学章．多元函数逼近．北京：科学出版社，1988．
[7] D. E. Macon, N. and Dupree. Existence and uniqueness of interpolating rational functions. The Amer Math.Monthly, 69: 751-759, 1962.
[8] 朱功勤．二元有理插值逐步算法．合肥工业大学学报(自然科学版)，1：10-15，2000．
[9] 朱晓临．(向量)有理函数插值的研究及其应用．中国科学技术大学：博士学位论文，2002．
[10] 朱功勤，顾传青．二元thiele型向量有理插值．计算数学，3：293-301，1990．
[11] 王仁宏，朱功勤．有理函数逼近及其应用．北京，科学出版社，2004．
[12] 王仁宏．数值有理逼近．上海，上海科学技术出版社，1980．
[13] L. Wuytack. on osculatory rational interpolation problem. Math.Comput., 29:837-843, 1975.
[14] P. Wynn. Continued fractions whose coefficients obey a non-commutative law of multiplacation. Arch. Rational Mech., 12: 273-312, 1963.
[15] P. R. Graves-Morris. Vector valued rational interpolants ⅰ. Number. Math., 42:331-348, 1983.
[16] P. R. Graves-Morris. Vector valued rational interpolants ⅱ. IMA.J.Numer.Anal., 4: 209-224, 1984.
[17] P.R. Graves-Morris and C.D. Jenkins. Vector valued rational interpolants ⅲ. Constr.Approx., 2: 263-289, 1986.
[18] 钟慧湘，王钲旋，庞云阶．图象重建中的有理逼近方法．中国图象图形学报，11：976-979．2000．
[19] 顾传青．矩阵切触有理插值．高等学校计算数学学报，2，1996．
[2] J．Stoer，R．Bulirsch，孙文瑜等译．数值分析引论．南京：南京大学出版社，1995．
[3] H. E. Salze. Note on osculatory rational imerpolation. Math. Computer, 19: 486-491, 1962.
[4] J. Stoer. Uber zwei algorithmen zur interpolation mit rationalen funktionen. Numerische Mathematik, 3: 285-304, 1961.
[5] 朱功勤，马锦锦．构造切触有理插值的一种方法．合肥工业大学学报，29，2006．
[6] 王仁宏，梁学章．多元函数逼近．北京：科学出版社，1988．
[7] D. E. Macon, N. and Dupree. Existence and uniqueness of interpolating rational functions. The Amer Math.Monthly, 69: 751-759, 1962.
[8] 朱功勤．二元有理插值逐步算法．合肥工业大学学报(自然科学版)，1：10-15，2000．
[9] 朱晓临．(向量)有理函数插值的研究及其应用．中国科学技术大学：博士学位论文，2002．
[10] 朱功勤，顾传青．二元thiele型向量有理插值．计算数学，3：293-301，1990．
[11] 王仁宏，朱功勤．有理函数逼近及其应用．北京，科学出版社，2004．
[12] 王仁宏．数值有理逼近．上海，上海科学技术出版社，1980．
[13] L. Wuytack. on osculatory rational interpolation problem. Math.Comput., 29:837-843, 1975.
[14] P. Wynn. Continued fractions whose coefficients obey a non-commutative law of multiplacation. Arch. Rational Mech., 12: 273-312, 1963.
[15] P. R. Graves-Morris. Vector valued rational interpolants ⅰ. Number. Math., 42:331-348, 1983.
[16] P. R. Graves-Morris. Vector valued rational interpolants ⅱ. IMA.J.Numer.Anal., 4: 209-224, 1984.
[17] P.R. Graves-Morris and C.D. Jenkins. Vector valued rational interpolants ⅲ. Constr.Approx., 2: 263-289, 1986.
[18] 钟慧湘，王钲旋，庞云阶．图象重建中的有理逼近方法．中国图象图形学报，11：976-979．2000．
[19] 顾传青．矩阵切触有理插值．高等学校计算数学学报，2，1996．