The Existence Of Cauchy Type Multivariate Rational Interpolation

It is well known that interpolation method is a classical method for approximation. The approximating expression is obtained by the function's values in finite points. Besides, a lot of numerical methods such as mechanical quadrature are obtained according to the interpolation method. The theories and methods of polynomial interpolation are very mature. Everyone knows that it is superior for rational than polynomial interpolation to arbitrary function. In 1821, the one variable rational interpolation was produced now called Cauchy interpolation. But in multivariate situation, we know a little, because that we even do not know how to express it. It is because that until 90's of 20 century, M.G.Marinari and H.M.Mo11er,T.mora given the constructive theory and algorithm of multivariate polynomial interpolation. Until then, the problem of the construction of interpolation polynomial was settled. On the light of this works, the constructive method of Cauchy type multivariate rational interpolation is given in this thesis, and we prove that it is classical Cauchy interpolation in one variable situation. Next, we also obtain the existence condition of the rational interpolating function.Given m + n + I inequality points x0,x1, ???,xm+n in real axis, and the corresponding function values f0, f1, ??? fm+n R of y = f(x), try to find a rational fractional functioncalled one variable rational interpolation.It is produced by Cauchy in 1821, so it is also called Cauchy interpolation.Given a set of inequality points V = {X0,X1, ???,XN} in s dimensional space, then there exists a ideal 1^ generated by polynomials which is zero in V, such that f(Xi) = 0,i = 0, ,N, for any f(X) IV. Let K[X] be the polynomial set with coefficients in K, then K[X] is a polynomial ring, K[X]/IV is its quotient ring. K[V]/Iv must be with dimension of N + 1 as K-vector space, because IV only has N + 1 zero points.Choose Lexicographic Order as monomial order in s dimensional space, i.e.: for any monomial Xa, X in K[x],the left-most nonzero entry is positive.If the basis of K-vector space are W0,W1,-,WN, and Wi w, then for any given offset y0, y1, , yN, polynomial interpolationexists and is unique.So, if the basis of K-vector space K[X]/IVare W0,W1,- , WN, we chooseFor given interpolation points X0,x1, XN, try to find a rational function with the form of (3) satisfy the interpolation conditionsare the Cauchy type multivariate rational interpolation's proper expression.For given inequality interpolation node points set is difficult to compute (also called coordinate function ring of V) directly. M.G.Marinari, H.M.Moller and T.mora note that the assignment of a fixed point defined a linear function in polynomial space, especialy when are inequality, linear function Lj(p(X)) - p(Xj) with assignment of Xj(j= 0,1, N) are linearly independent. They generate a set of basis of dual space L of K[X]/IV, we can compute the dual basis of K[X]/Iv, and the algorithm is as follows.Algorithm: For given point Xj, define a linear functiondenote, be the monomial sequence in 5 dimensional space with Lexicographic Order. Consider sequenceFrom definitionwe choose the front N + 1 linear inequality vector in (5), suppose they are:then are the coordinate function ring's basis corresponding to V Write as Construct matrixderived from B(X, y} after erasing the last line, is (N +1) (N + 2) matrix. And letObviously B = (B1, B2), and RandB1 =m + 1.Now write interpolation conditionas(6)it is obviously that there existing which are not all equal to 0 is equivalence todet B(X, y) = 0, (7)or,Upper expression is a polynomial about X, y expanded from B(X,y), and B1j is the matrix erasing the column of B1 which include Wj in B, B2j is the matrix erasing the column of B2 which include Wj in B.So when det B(X, y)0, we obtainTheoreml: For given inequality interpolation node points, and the offset Cauchy type multivariate interpolationhas solution is equivalen...