| Interpolation is an important method in numerical approximation. Rational func-tions sometimes are superior to polynomials for interpolating data because they can achieve more accurate approximations with the same amount of computation. In ad-dition, rational interpolants have a natural way of interpolating poles whereas polyno-mial interpolants do not. So how to solve the problem of rational interpolation is what people have been concerned with. In this paper we used the theory of algebraic geom-etry to study the multivariate Cauchy rational interpolation, multivariate osculatory rational interpolation, vector-vector osculatory rational interpolation and polynomial interpolation. The main results of this paper are as follows:In Chapter2we present the Fitzpatrick algorithm for Cauchy rational interpola-tion.Multivariate Cauchy rational interpolation:Given a set of L distinct points{Y1,…, YL} in space Kn. At point Yi, the function value is prescribed as fi∈K. The multivariate Cauchy rational interpolation problem is to find a rational interpolation function such that r(Yi)=r(X)|Yi=fi,i=1,...,L, where a∈P=K[X], b∈P, b(Yi)≠0for all i.Definition1A pair (a, b) G V2is called a weak interpolation tuple for the multivariate Cauchy rational interpolation problem if a=bfi mod Li, i=1,...,L, where Ii=(X-Yi).Define (a,b)+(c, d)=(a+c,b+d), g(a,b)=(ga, gb). Then M={(a,b): a=bhi mod Ii,si-1, i=1,...,L}(?)P2is a P-submodule.Definition2(a term order (?)ζ on P2)(1) We say Xα(1,0)(?)ζ-Xβ(1,0) if and only if|α|<|β|, or|α|=|β|and Xα(?)lex Xβ;(2) We say Xα(1,0)(?)ζ Xβ(0,1)if and only if|α|≤|β|+ζ;^(3) We say Xα(0,1)(?)ζXβ(0,1) if and only if|α|<|β|,or|α|=|β|and Xα(?)lex Xβ; where (?)lex is the lexicographic order on V, and ζ is a given integer.It is easy to check that the order (?)ζ is a term order on V2.Fix a term order (?)ζ. We can apply the Fitzpatrick algorithm to compute the Grobner basis of the submodule M. If {(a1,b1),...,(amL,bmL)} is a Grobner basis of the submodule M, then any pair (a, b) with the form of (a, b)=c1(a1,b1)+...+ct(amL,bmL) is a weak interpolation tuple, where cj∈P (j=1,...,mL) are free parameters. Choose Cj properly such that b(Yi)≠0, i=1,...,L, then we can get the interpolation function So we get the parametric solution of all the interpolation functions with the given complexity. Based on the Fitzpatrick algorithm, we present a Fitzpatrick-Neville type algorithm for Cauchy interpolation. With this algorithm, we can determine the value of the interpolating function at a single point without computing the rational interpolating function. In Chapter3we present the Fitzpatrick algorithm for multivariate osculatory rational interpolation. Based on the Fitzpatrick algorithm and the properties of an Hermite interpolation basis, we present a Fitzpatrick-Neville type algorithm for mul-tivariate osculatory rational interpolation. It may be used to compute the values of osculatory rational interpolants at some points directly without computing the rational interpolation function explicitly.Multivariate osculation rational interpolation:Given a set of L distinct points{Y1,...,YL} in space Kn. At point Yi, the values of the derivatives Dα are prescribed as fi(α)∈K for a in the lower set Ai. The multivariate osculatory rational interpolation problem is to find a rational interpolation function such that Dαr(Yi)=Dar(X)|Yi=fi(α),(?)α∈Ai,i=1,...,L, where α∈P,b∈P, b(Yi)≠0for all i.From Dαr(Yi)=fi(α), for each α∈Ai, i=1,...,L, we can obtain the equiv alent definition of multivariate osculatory rational interpolation as:the Taylor series expansion of r(X) at the point X=Yi satisfiesLet si=#Ai, i=1,...,L,N=(?). Let Ii,si-1={p∈P:Dαp(Yi)=0,(?)α∈Ai}. For each point Yi and the corresponding lower set Ai, define polynomial hi Definition3A pair (a, b)∈P2is called a weak interpolation tuple for the multivariate osculatory rational interpolation problem if a=bhi mod mod Ii,si-1,i=1,...,L. Define (a, b)+(c,d)=(a+c,b+d), g(a, b)=(ga, gb). Then M={(a,b):a=bhi mod Ii,si-i,i=1,..., L}(?)P2is a P-submodule.So we can apply the Fitzpatrick algorithm to compute multivariate osculatory rational interpolation, and get the parametric solution of all the interpolation functions with the given complexity. Based on the Fitzpatrick algorithm and the properties of an Hermite interpolation basis, we present a Fitzpatrick-Neville type algorithm for multivariate osculatory rational interpolation. It may be used to compute the values of osculatory rational interpolants at some points directly without computing the rational interpolation function explicitly.Vector-valued rational interpolation is a natural generation of rational interpola-tion. In Chapter4we first apply the Fitzpatrick algorithm to multivariate vector-valued osculatory rational interpolation. Then based on the Fitzpatrick algorithm and the properties of an Hermite interpolation basis, we present a Fitzpatrick-Neville-type al-gorithm for multivariate vector-valued osculatory rational interpolation. It may be used to compute the values of multivariate vector-valued osculatory rational interpolants at some points directly without computing the interpolation function explicitly.Polynomial interpolation can be treated as a particular rational interpolation. In Chapter5we derive a variation of the Fitzpatrick algorithm, which can be used to determine not only the function value but also the differential values of the Hermite interpolation polynomial at a single point directly without computing the interpolation polynomial. 21hik∈P, I(k)P]¨ a.2002ìO'Keefe Fitzpatrick[85 |¤′′Input: M±Gro¨bner G=...
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