Some Studies On Cubic Lagrange Polynomial Interpolation On The Rectangle

Multivariate polynomial Lagrange interpolation is a hot problem for discussion in the field of multivariate approximation.It has more complex characters than those of univariate interpolation.When we interpolate on multivariate polynomial space,the problem which at first we need to resolve is the properly interpolating,another important problem is constructing interpolation formula.In this paper,we just research in these two problems and obtain some corresponding results.Firstly,the background and the main results of multi-Lagrange interpolation are introduced in the first chapter,and in the introduction,we focus on describing the results of construction of properly posed set of nodes(or PPSN,for short) on R~2.Secondly,we research deeply the properly interpolating problem about set of nodes on rectangular,and then obtain corresponding conclusions which are about posedness of three Lagrange interpolation and basis functions' construction.In this paper,we choose 12 points:the middle point and the two points that is symmetrical about the middle point on every edge.How do we choose the ten points from the given 12 points so that the ten points construct PPSN? For this problem, we firstly get the law how we choose the PPSN from the complete numerical calculation,and then prove it by using the superposition interpolation method.Then,for these PPSN,basing on the idea of superposition interpolation and three elliptical of Chapterâ…¢,we construct the corresponding basis function and interpolation formula.At last,we choose different PPSN and different interpolated function,do lot of numerical experiments to test the error,then obtain some conclusions,it is not matter for the error that choosing different PPSN,for example.Furthermore,we research related issues on general quadrilateral and obtain similarities and differences compared to rectangle and the future work.