向量值有理插值函数的递推算法  

A Recurrence Algorithm for Vector Valued Rational Interpolation

作　　者：朱晓临[1] 朱功勤[2] 

机构地区：[1]中国科学技术大学数学系,合肥230026 [2]合肥工业大学理学院,合肥230009

出　　处：《中国科学技术大学学报》2003年第1期15-25,共11页JUSTC

基　　金：国家自然科学基金资助项目 (No .10 1710 2 6)

摘　　要：针对向量连分式序列Rn(x) =b0 + x-x0b1 +… + x-xn- 1 bn ,n =0 ,1 ,2 ,…利用向量的Samelson逆 ,建立了类似于标量逐步有理插值算法的向量有理函数插值的逐步递推算法 :Pλ =dλ,λPλ- 1 + ∑λ-1i=1wλidλ-i,λPλ-i- 1 + (x -xλ- 1 ) 2 Pλ- 2 +ωλλBλ,Qλ =dλ,λQλ- 1 + ∑λ-1i=1wλidλ-i,λQλ-i- 1 + (x-xλ- 1 ) 2 Qλ- 2 ,　λ=2 ,3,… ,n( )其中 P0 =b0 ,Q0 =1 ;　　 P1 =d1 ,1 P0 +ω1 1 b 1 ,Q1 =d1 ,1 Q0 ,Rλ(x) =Pλ(x)Qλ(x) (λ=0 ,1 ,… ,n)是满足插值条件Rλ(xi) =Rλ(xi)Qλ(xi) =Vi,i =0 ,1 ,… ,λ的向量有理函数 与向量有理函数插值的传统算法相比 ,上述算法的主要优点是具有承袭性 :当需要增加一个插值条件Rn+1 (xn+1 ) =Vn+1 时 ,原来已经得到的向量有理插值函数序列 P0Q0 ,P1 Q1 ,… ,PnQn 仍然保留 ,只要按 ( )式再计算一个Pn+1 (x) ,Qn+1 (x)即可 .在此基础上 ,将上述算法推广到二元情形 。Based on the vector valued continued fractionsR n(x)=b 0+x-x 0b 1+...+x-x n-1 b n, n=0,1,2,...a recurrence algorithm for vector valued rational interpolation, which is simila r to the scalar recurrence algorithm with inheritance, is established as follows by use of Samelson inverse:P λ=d λ,λ P λ-1 +∑λ-1i=1w λ id λ-i,λ P λ-i-1 +(x-x λ-1 ) 2P λ-2 +ω λ λB λ, Q λ=d λ,λ Q λ-1 +∑λ-1i=1w λ id λ-i,λ Q λ-i-1 +(x-x λ-1 ) 2Q λ-2 , λ=2,3,...,n(*) \%where\%P 0=b 0, Q 0=1; P 1=d 1,1 P 0+ω 1 1b * 1 , Q 1=d 1,1 Q 0,R λ(x)=P λ(x)Q λ(x)(λ=0,1,...,n) are the vector valued rational interpolants which satisfy the interpolating c onditionsR λ(x i)=R λ(x i)Q λ(x i)=V i, i=0,1,...,λAs compared with the classical algorithms for the vector valued rational interpo lation, the inheritance is the main advantage of the algorithm presented in this paper. That is, if we want to increase a new interpotating andition R n+1 (x n+1 )=V n+1 , we only need to computate P n+1 (x), Q n+1 (x) by (*), and the original vector value d rational interplants P 0Q 0, P 1Q 1,...,P nQ n stay unchanged. In addi tion, the above algorithm is generalized to the bivariate case and some numerica l examples are presented to illustrate the efficiency of this algorithm.

关 键 词：向量连分式 递推算法 向量有理插值 

分 类 号：O241.5[理学—计算数学]