构造切触有理插值的一种方法

第29卷第10期　2006年10月合肥工业大学学报(自然科学版)J OU RNAL OF H EFEI UN IV ERSIT Y OF TECHNOLO GYVol.29No.10　Oct.2006　收稿日期:2005210219;修改日期:2005212209作者简介:朱功勤(1938-),男,安徽萧县人,合肥工业大学教授,博士生导师.构造切触有理插值的一种方法朱功勤,　马锦锦(合肥工业大学理学院,安徽合肥　230009)摘　要:切触有理插值是Hermite 插值的一种推广,已有的构造切触有理插值方法都与连分式相联系,因此其算法可行性是有条件的,且计算量较大,讨论无条件的构造切触有理插值的方法具有实际应用价值。

利用凸组合方法可方便地构造出数量值切触有理插值函数或向量值和矩阵值函数,其构造过程公式化,便于在计算机上实现,且计算量较小,具有广阔的应用前景。

关键词:切触插值;凸组合方法;插值公式;参数.中图分类号:O241.5 文献标识码:A 文章编号:100325060(2006)1021320204A w ay of constructing osculatory rational interpolationZHU G ong 2qin ,　MA Jin 2jin(School of Sciences ,Hefei University of Technology ,Hefei 230009,China )Abstract :Osculatory ratio nal interpolation is a generalization of Hermite interpolation.As t he existing met hods of co nstructing o sculatory rational interpolation are all related to continued f ractions ,t he fea 2sibility of t heir algorit hms is conditional and t hey need a large amount of calculation.In t his paper ,t he met hod of convex combination is used to construct t he osculatory rational interpolating f unction.The p resented met hod can also be used to const ruct conveniently t he vector 2valued osculatory rational interpolating f unction or t he mat rix 2valued o sculatory rational interpolating f unction.The course of const ructing is formulary and convenient to realize on t he comp uter ,and it needs a small amount of calculation ,so t he p resented met hod has a bright application f ut ure.K ey w ords :o sculatory interpolation ;met hod of convex co mbination ;interpolating formula ;parameter 设x 0<x 1<…<x n ,所谓切触有理插值问题,就是寻求有理分式函数p (x )/q (x ),使得d d x kp (x )q (x )x =x i=f (k )i(k =0,1,…,s i -1;i =0,1,…,n )(1)其中,p (x )、q (x )为实系数多项式;f i =f (x i )。

文献[1,2]利用连分式方法给出了构造切触有理插值的递推算法,这些算法可以称之为经典的方法。

文献[3]利用代数方法证明了切触有理插值问题有解的充要条件;文献[4]也给出了一种构造切触有理插值的方法。

上述结果虽然很好,但不便于实际应用。

所谓向量值切触有理插值问题,就是寻求向量值有理分式,即N (x )Q (x )=(n 1(x ),…,n d (x ))Q (x )使得d d xkN (x )Q (x )x =xi=V (k )i(k =0,1,…,s i =1;i =0,1,…,n )(2)其中,Q (x )和n j (x )(j =1,…,d )是实系数多项式,记M =6ni =0s i -1。

文献[5]利用连分式及Samelson 逆给出构造向量值切触有理插值函数的方法,但算法可行性是有条件的,且计算量较大。

本文引入有理基函数,利用凸组合[6]方法,给出一种构造切触有理插值函数的方法。

为简化计算,针对s i =2(i =0,1,…,n )进行讨论。

1　预备知识对于给定的x 0<x 1<…<x n ,记w i (x )=(x -x 0)…(x -x i-1)(x -x i+1)…(x -x n )(3)它是右端不含因子(x -x i )的n 次多项式。

因此[w i (x )]2是2n 次多项式,记D =dd x。

容易看出w i (x l )=0,i ≠l (i ,l =0,1,…,n ),而w i (x i )=D w i (x )x =x i )定义有理分式函数如下,即βi (x )=[w i (x )]2Q (x )(4)其中Q (x )=[w 0(x )]2+[w 1(x )]2+…+[w n (x )]2=6ni =0[w i (x )]2(5)直接验证可知βi (x )具有如下性质:(1)βi (x )是有理分式函数,即βi (x )∈R (2n ,2n ),分子、分母都是2n 次多项式;(2)βi (x l )=1　i =l 0　i ≠l　(i ,l =0,1,…,n );(3)D βi (x l )=0　(i ,l =0,1,…,n )。

事实上D βi (x )=1[Q (x l )]2{Q (x )2w i (x )w ′i (x )-[w i (x )]2[2w 0(x )w ′0(x )+2w 1(x )w ′1(x )+…+2w n (x )×w ′n (x )]}而 Q (x l )=[w l (x l )2]≠0于是D βi (x l )=1[Q (x l )]2{[w l (x l )]22w i (x l )×w ′i (x l )-[w i (x l )]2[2w 0(x l )×w ′0(x l )+…+2w n (x l )w ′n (x l )]}当i ≠l 时D βi (x l )=1[Q (x l )]2{[w l (x l )2・0-0・[2w l (x l )w ′l (x l )]}=0当i =l 时D βi (x i )=1[Q (x i )]2{[w i (x i )]22w i (x i )×w ′i (x i )-[w i (x i )]2[0+…+2w i (x i )w ′i (x i )+…+0]}=02　切触有理插值公式先讨论数量切触有理插值问题。

设给定x 0<x 1<…<x n 及相应函数值f (x i )和导数值f ′(x i )(i =0,1,…,n ),引入插值算子如下:p i (x )=(x -x i )f ′(x i )+f (x i )(i =0,1,…,n )(6)显然p i (x i )=f (x i ),p ′(x i )=f ′(x i )(i =0,1,…,n )。

利用插值算子p i (x )及βi (x )作线性组合,即R (x )=6ni =0βi(x )p i(x )(7)(7)式为数量切触有理插值公式。

由βi (x )的性质(2)可知R (x l )=6ni =0βi(x l)p i(x l)=βl (x l )p l (x l )=p l (x l )=f (x l )(l =0,1,…,n )R ′(x )=6ni =0β′i (x )p i (x )+βi (x )p ′i (x )由βi (x )的性质(3)及(6)式可知R ′(x l )=6ni =0[β′i(x l)p i(x l)+βi (x l )p ′i (x l )]=βl (x l )p ′l (x l )=p ′l (x l )=f ′(x l )(l =0,1,…,n )易看出R (x )∈R (2n +1,2n )。

例1　x 0=0,x 1=1,f (0)=1,f ′(0)=2,f (1)=2,f ′(1)=3。

试求满足插值条件的有理函数。

解　由(6)式知插值算子为p 0(x )=2x +11231　第10期朱功勤,等:构造切触有理插值的一种方法p 1(x )=3x -1由(4)式知β0(x )=(x -1)2(x -1)2+x 2β1(x )=x2(x -1)2+x 2由(7)式得R (x )=61i =0βi(x )p i(x )=(x -1)2p 0(x )+x 2p 1(x )(x -1)2+x2=5x 3-4x 2+12x 2-2x +1直接验证知R (x i )=f (x i ),R ′(x i )=f ′(x i ),i =0,1。

值得指出的是:R (x )分子、分母多项式次数比用连分式方法所得的结果R (x )=5x 2-x -23x -2高一次。

但可采用下面方法将分子或分母多项式降低1次。

引入参数α0、α1,考虑有理分式函数R (x )=α0(x -1)2p 0(x )+α1x 2p 1(x )α0(x -1)2+α1x 2(8) 利用2多项式相等的充要条件,选择参数α0、α1,使分子多项式或分母多项式次数降低1次。

以降低分子多项式次数为例,选择参数α0、α1,使2α0+3α1=0,即分子多项式中x 3系数为0,将p 0(x )、p 1(x )及α0、α1代入(8)式得R (x )=7x 2-3-x 2+6x -3虽然用(7)式构造出的有理分式函数次数高一点,但由于构造方法简单,在实际应用时还是可取的。

下面重点讨论向量值切触有理插值函数的构造问题。

对给定的x 0<x 1<…<x n 及相应的向量V (x i )和导数值V ′(x i )(i =0,1,…,n ),定义n +1个插值算子。