Study On Multivariate Interpolation Of Saddle Surfaces

In the late twentieth century,multivariate Lagrange interpolation began to develop rapidly.In 1965,Professor Liang Xuezhang began to study this kind of interpolation in his master's dissertation "Interpolation and Approximation of Multivariate Functions".For the first time,the regularity of interpolation was transformed into geometric problem,and a geometric method was proposed to study the existence and uniqueness of interpolation node groups.Up to now,the research on multivariate interpolation is mainly divided into two directions: one is that the interpolation space has been determined to construct the corresponding regular node group;the other is that the determined node group has been given to construct the corresponding regular interpolation space,and the number of times of constructing the interpolation space should be as low as possible.This article will introduce this paper from three parts.Chapter 1 is the basic theory of multivariate polynomial interpolation.Aiming at the idea of univariate function interpolation,which is mainly Lagrange interpolation,the bivariate function interpolation is described.We extend the method of constructing basis function to binary function,discuss the interpolation problem of binary function,and mainly discuss the problem of well-posed node group of binary function interpolation.Chapter 2 introduces the problem of ternary interpolation,which is mainly divided into two parts.In the first section,the basic concepts of ternary interpolation problems are introduced.In the second section,the basic theory of the set of well-posed nodes along space algebraic surfaces and algebraic curves is introduced.In the second section,we mainly introduce four methods to construct the set of suitable nodes for interpolation,namely "adding straight line method","adding plane method","adding algebraic curve method" and "adding algebraic surface method".The third chapter is the main content of this paper,taking saddle surface as an example to analyze.On the basis of the existing research results,the saddle surface method is proposed to construct the set of interpolation nodes,and the feasibility of the method is proved by an example in Matlab software.