Some Research On Rational Interpolation

As a generalization of polynomial interpolation, rational interpolation has many advantages. In this paper we study matrix valued rational interpolation problem and Birkhoff-type rational interpolation.â… . Matrix Valued Rational InterpolationIn this paper we convert the task of seeking the weak solution (P(X), q(X)) of matrix valued osculatory rational interpolation into computing the Groebner bases of R-submodule M of the free module over the polynomial ring. This Groebner bases can be computed recursively. Then we can obtain a parametric rational interpolation function with the Groebner bases. By choosing these parameters properly, we may get the desired matrix valued rational interpolation function.A subsetΑ(?) Nⁿ is called delta set, if it closed under the divison order; that is, ifα∈Αthenβ∈Αfor 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 {Α₁,... ,Α_L}. And corresponding function values (?). Construct a rational interpolation functions.tFrom the definition of matrix valued oscillatory rational interpolation we know that the equivalent definition is: the Taylor series expansion of R(X) about the points X = Y_i satisfiesDefinition 2.For each point Y_i and the lower setΑ_i, define polynomial matrix H_i(X)that isDefinition 3.(weak interpolations) (P,q)∈(R^{d₁×d₂}[X],R[X]) is called a weak interpolation. ifWhere (?) is the vanishing ideal ofΑ_i.Let M = {(P,q)|P≡qH modâ… } (?) (R^{d₁×d₂}[X],R[X]) where H∈R^{d₁×d₂}[X] is a polynomial matrix, I (?) R[X] is an ideal. We defined (P₁,q₁) + (P₂,q₂) = (P₁ + P₂,q₁+q₂),g·(P₁,q₁) = (gP₁,gq₁). Thus M is a R-submoduleof (R^{d₁×d₂}[X],R[X]).Definition 4. (order (?)) 1. We say (?) if |α| > |β| or |α| = |β| and k₂ < k₁ , or |α| = |β| , k₁ = k₂ and l₂ < l₁ or |α| = |β| , k₁= k₂, l₂ = l₁ and (?);2. We say (?) if |α| > |β| +ξ;3. (?) if |α| > |β| or |α| = |β| and (?).where (?) is the lexicographic order on R[X], andξis a given integer.We can check that the order (?) is a monomial order on R^d+1.Definition 5.Let (?) = {{Y₁,Α₁,H₁),..., (Y_L,Α_L,H_L)}, We defined(?) is called matrix valued interpolation set, M(?) is called the matrix valued interpolation modules of (?).Proposition 1. Fix an order (?). Let M (?) (R^{d₁×d₂}[X],R[X]) be the submodule. We will denote by〈LT(M)〉the monomial submodule generated by the leading terms of all (?)∈M with respect to (?), then1. for any (?)∈(R^{d₁×d₂}[X],R[X]), there exists a unique element (?)∈(R^{d₁×d₂}[X],R[X]), s.t. (?) -(?)∈M, and (?) is a R-linear combination of monomials in (?).2. if (?) is finite-dimensional, thentherefore, dim (?). Where (?). Theorem 2. Fox an order on (R^{d₁×d₂}[X],R[X). Let W ={(Y₁,Α₁,H₁), ...,(Y_L,Α_L,H_L)} be the matrix valued interpolation set. Let V ={(Y₁,Α₁), ...,(Y_L,Α_L)}, I(V) be the vanishing ideal, thenProposition 3. M(?) is the matrix valued interpolation modules of (?). if (?), and (?), then (?) is a Groebner bases for M(?).Let M₀=(R^{d₁×d₂}[X],R[X]),(?),m=1.....N.Where (?) is the vanishing ideal ofΑ_i,H_i(X) is polynomial matrix.Obviously M = M_N and (?).We compute the Groebner bases of M_N recursively. It is easy to see that (?)is the Groebner bases of M₀. Using M₀, if we can compute the Groebner bases of M₁, the Groebner bases of M₂ recursively. We can compute the Groebner bases of M_N.If we have computed the Groebner bases of (?).Thuswhere (?) are the parameters.â…¡. Birkhoff-type rational interpolationThe problem of Hermite-Birkhoff rational interpolation.