Application Of Bell Polynomial To Inverting A Confluent Vandermonde Matrix

In this paper,we mainly study the problem of how to solve the inverse of a confluent vandermonde matrix,which can be solved by many methods.We use a method to solve the inverse of a confluent vandermonde matrix which is Bell polynomial.As we know that the confluent vandermonde matrix is closely related to the Hermite interpolation.A system of equations that satisfies the polynomial of Hermite interpolation,the equations of the undetermined coefficient matrix is the confluent of vandermonde matrix.It is naturally suggested that the inverse of the confluent vandermonde matrix can represent Hermite interpolation polynomial.There have been a lot of results about Hermite interpolation polynomials.This paper mainly considers the use of Hermite interpolation base function to represent the inverse of the confluent vandermonde matrix.Because the Hermite interpolation problem is a kind of interpolation with derivative conditions,It's an expression that involves the derivative of the high order of the compound function.We use a combinatorial mathematical tool which called Faà di Bruno's formula and Bell polynomials to represent Hermite interpolation polynomials.The organization structure of this paper is as follows.First we introduce the definition of the confluent vandermonde matrix and Faà di Bruno's formula and Bell polynomial.And then,we introduce Hermite interpolation.What's more,we introduce two important lemmas.Lemma3.1.Suppose that the function g(x)has all derivatives up to ordern at the point x then(?)Lemma3.2.Assume that x1,…,xs are group of distinct points,and ?1,…,?s are integers.(1)forx?x1,…,xs,the following equation hold(?)(2)forx = xi,1?i?s,for instance x?x1,if ?1?1,then the following equation holds(?)Those lemmas play an important role in solving the problems of this paper.Moreover we discuss the inverse of a confluent vandermonde matrix and give an example to verify the result of this paper.