Polynomial Interpolation In Several Variables

Polynomial interpolation in several variables is a subject which is currently an active area of research. First, we introduce and discuss the various methods of multivariate polynomial interpolation in the literature. Based on this study, we state multivariate Lagrange interpolation over again from algebraic geometry viewpoint:Given different interpolation nodes A1,A2 .....,An in the affine n-dimensional space Kn, and accordingly function values fi(i = 1,..., m), the question is how to find a polynomial p(X) K[x1, x2,...,xn] satisfying the interpolation conditions:where X= (x1,X2,....,xn). Similarly with univariate problem, we have provedTheorem If the monomial ordering is given, a minimal ordering polynomial satisfying conditions (1) is uniquely exsisted.Such a polynomial can be computed by the Lagrange-Hermite interpolation algorithm introduced in chapter 2. Another statement for Lagrange interpolation problem is:Given monomials 1 ,2 ,.....,m from low degree to high one with respect to the ordering, some arbitrary values fi(i= 1,..., m), find a polynomial p(X), such thatIf there uniquely exists such an interpolation polynomial p{X), the interpolation problem is called properly posed. Because multivariate polynomial interpolation is not the simple generalization of univariate context, one must solve the poisedness of interpolation problems first of all. Hereon, we have cited a very practicable principle for judging the properly posed set of nodes by means of Grobner bases theory.How to construct Lagrange interpolation polynomial is quite important. Now the algorithm introduced in chapter 2 is used widely, in the paper, we obtain p(X) by computing interpolation bases 1,2,.....,m and carry it out using programme CoCoA. The examples given in the end show that our method is very simple and efficient.The stability of interpolation problems is much interesting, however, the research about it is poor. So we discuss this character primarily, and cite several examples to make out its complexity.