The Construction Of PPSN And Interpolation Bases On Multivariate Lagrange Interpolation

Polynomial interpolation in several variables is a subjectwhich is currently an active area of research. First, we introduce anddiscuss the various methods of multivariate polynomial interpolationin the literature.In particular,using the multivariate divided differenceintroduce by C.de Boor,we obtain a error formula for tensor productinterpolation in R~s.Multivariate polynomial interpolation is not the simple generaliza-tion of univariate context.First of all,one must solve the well-posedproblem.One of the work is Liang's theorem,which transfer the bi-variate Lagrange interpolation into a geometrical problem by meansof algebraic curves.In chapter three,we discuss Chung and Yao's GCcondition and improve Liang's theorem.Furthermore,using the resultof variety in algebraic geometry, we study the geometrical structure ofproperly posed set of nodes(PPSN) for interpolation on algebraic hy-perplane, and give a Hyperplane Superposition Process to constructthe PPSN for interpolation on algebraic hyperplane,therefore we makeclear the geometrical structure of PPSN for multivariate Lagrange in-terpolation.The base calculating multivariate Lagrange interpolation with al-gebraic method since GrSbner bases suggesting that, can be usedbecomes possibility.In chapter four,we give out the algorithm thatGrobner method and CoCoA programming method calculating multi-variate Lagrange interpolation bases and several examples. The formof the multivariate Lagrange interpolation bases on PPSN is given.Theform of multivariate Lagrange interpolation bases on unproperly posedof nodes is more complex. The research about it is poor.So we dis- cuss this character primarily, and cite several examples to make outits complexity.