The Reaserch Of Multivariate Graded Interpolation

Content:Multivariate interpolation is a very important reaserch field in compulational mathematic. This is a both basic and classic mathematical problem. It is an important part in compulational mathematic. So far people get many research results about multivariate interpolation. Because multivariate graded interpolation has awidespread application in many field, people have come to focus on multivariate graded interpolation. We reseach further multivariate interpolation and multivariate graded interpolation in this paper. And this paper gives new constructive methods of properly posed set of nodes in bivariate graded interpolation.This paper is composed of three charpters. In the first charpter, we introduce the theory and constructive methode of multivariate polynomial interpolation and multivariate interpolation on algebraic vareieties. We concult known results. In the second charpter, we talk about bivariate Lagrange interpolation along an algebraic curve and multivariate Lagrange interpolation along an algebraic surface and an algebraic curve in R 3. We crstallize this problem in this charpter and give some result on multivariate interpolation. In the third charpter, on the basic of the theories of bivariate polynomial interpolation of total degree, we give new constructive methods of properly posed set ofnodes in bivariate graded interpolation: every degree algebraic curve process. Theorem3.2.1 {Q_i}_i=1^k is a interpolation properly posed set of nodes about interpolation space Pm , n in R 2. If every point in{Q_i}_i=1^kis not on curve y=x², m + 2 n+5 different points in this curve and {Q_i}_i=1^kis a interpolation properly posed set of nodes about Pm + 2 ,n+1. On the basic of this theorem, we get following theorem: Theorem3.2.2 {Q_i}_i=1^kis a interpolation properly posed set of nodes about interpolation space Pm , n in R 2. If every point in{Q_i}_i=1^kis not on curve y= a₀+a₁x+a_tx^t,(at≠0), m+tn+2t+1 different points in this curve and{Q_i}_i=1^kis a interpolation properly posed set of nodes about P_m+t,n+1.