An Algorithm Of Osculatory Rational Interpolation Based On R-modules

In this paper,we analyze the Patrick algorithms for (multivariate) osculatory rational interpolation. Using Hermite interpolation polynomial and the Groebner bases for vanishing ideals,we structure the Groebner bases for a R-modules with the order (?). So we can compute the Groebner bases for R-modules with the order (?), using Buchberger' algorithm for submodules. So we can get a parametrization of multivariate oscillatory rational function a(X)/b(X) interpolating a given set. And for the rational interpolation problem on the interpolation set W = (?),we proof that the dimension of the quotient module equal numbers of interpolation conditions. Taking into account the vector valued osculatory rational interpolation problem is the generalization of the osculatory rational interpolation problem. We convert the vector valued osculatory rational interpolation problem into computing the Groebner bases for a submodules. And for the rational interpolation problem on the interpolation set (?), we also proof that the dimension of the quotient module equal numbers of interpolation conditions.And we can give a Newton-type algorithm which can be use to compute a parametrization of vector valued multivariate osculatory rational function a(X)/b(X).A subset (?) is called delta set, if it closed under the divison order; that is, ifα∈A thenβ∈A for allα= (α₁,…,α_n)β= (β₁,…,β_n) componentwise.Let D a differential operator.The problem of osculatory rational interpolation can be stated as follows:Definition 1. Given a set of L distinct points in space Rⁿ {Y₁,…,Y_L}. Each point have multiplicity defined by the sets {A₁,…,A_L}. And corresponding function values (?). Construct a rational interpola- tion functionDefinition 2.(weak interpolations)Let h_i(X) be the polynomial interpolating the interpolation point Y_i, a pair (a, b)∈R² is called a weak interpolation of the set of polynomials h_i ifWhere I_i,si-1) is the vanishing ideal of A_i.Let M = {(a, b)|a = bh_i mod I_i,si-1),i = 1,…,L}, we defined (a, b) -(c, d) = (a + c, b + d),g ? (a, b) = (ga, gb). Thus M is a R-submodule of R². Let H(X) be Hermite interpolation polynomial and G = {g₁,…,g_t} be the Groebner bases for vanishing ideals I = (?).Theorem 1.(?) be the Groebner bases for M with the order (?).So we can compute the Groebner bases for R-modules with the order (?),using Buchbcrger' algorithm for submodulcs.Definition 3. Let W = (?), whereÏ„_i∈K, A_i∈Nⁿ, h_i(X) is the polynomial interpolating the interpolate pointÏ„_i. We definedW is called interpolation set, M(W) is called the interpolation modules of W.Theorem 2. Fox an order on R². Let (?) be the interpolation set. Let (?) be the vanishing ideal, thendim (R²/M(W)) = #N(M(W)) =(?)Taking into account the vector valued osculatory rational interpolation problem is the generalization of the osculatory rational interpolation problem. We convert the vector valued osculatory rational interpolation problem into computing the Groebner bases for a submodules. And for the rational interpolation problem on the interpolation set (?), we also proof that the dimension of the quotient module equal numbers of interpolation conditions.And we can give a Newton-type algorithm which can be use to compute a parametrization of vector valued multivariate osculatory rational function a(X)/b(X).Definition 4. Given a set of L distinct points in space Rⁿ {Y₁,…,Y_L}. Each point have multiplicity defined by the sets {A₁,…,A_L}. And corresponding vector valued (?). Construct a vector valued rational interpolation functions.tDefinition 5.(weak interpolations) A pair (?) is called a weak interpolation of the set of polynomials (?) ifWhere (?) mod (?).Definition 6. Let (?), We defined (?) is called vector valued interpolation set, (?) is called the vector valued interpolation modules of (?).Theorem 3. Fox an order on R^d+1. Let (?) be the vector valued interpolation set. Let (?) be the vanishing ideal, thenUsing Theorem 3, we can give a Newton-type algorithm which can be use to compute a parametrization of vector valued multivariate osculatory rational function (?)(X)/b(X).