Research On Construction Methods And Existence Of Rational Interpolants Function

Rational function is simple function class.Compared to polynomial interpolations,it is more flexible and can describe physical character of functions more accurately, approximation effect is better.Therefore it has widely used in numerical approximation,function approximation,etc.Because rational interpolation is an important aspect of rationalapproximation.It is very much concerned about research on theory and method of rational interpolation. However , method of construction rational interpolation function is related the number of the type of rational function .It can give solvability conditions for interpolation problem by use of construction. In this thesis , on the research foundation ,the use of difference quotient give a method of judging the solvability problem of rational interpolation, and we extend it to vector-valued rational interpolation problem ,obtain corresponding conclusions.In this thesis , Frist ,we introduce the background of the rational interpolation , the basic knowledge and method of rational interpolation briefly,as well as the major work of this article.In chapter 2, we studied method of construction rational interpolation with Newton's difference quotient.By use of difference quotient we give a solvability condition of rational interpolation problem which reduce computation ,and extend it to bivariate rational interpolation and vector-valued rational interpolation function.In chapter 3,we discuss conditions of Thiele, Newton-Thiele and Thiele-Newton type rational interpolation function for the existence, and give specific examples to verify.In chapter 4,we introduce a rational interpolation formula which is as close as possible concrete, reveal the internal structure of bivariate rational interpolation. we can get a necessary and sufficient conditions of bivariate rational interpolation function for the existence, give an expression of the formula that is concrete,and extend it to the vector.