A Newton formula for generalized Berzolari-Radon sets
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A Newton formula for generalized Berzolari-Radon sets J. M. Carnicer · C. God´es
Received: 5 July 2013 / Accepted: 8 May 2014 © Springer Science+Business Media New York 2014
Abstract We introduce an extension of the Newton formula for bivariate generalized Berzolari-Radon sets and suggest a generalization of the divided differences for that kind of sets. We apply this formula to two bivariate problems where the nodes are distributed on lines or concentric circles respectively. Some examples are provided. Keywords Bivariate polynomial interpolation · Berzolari-Radon set · Newton formula Mathematics Subject Classifications (2010) 41A05 · 41A63 · 65D05
1 Introduction The Berzolari-Radon construction [1, 9] in the plane makes it possible to reduce an interpolation problem of degree n in 2 dimensions into two subproblems: a usual univariate problem and a bivariate problem of degree n − 1. It is based on a general observation (cf. [7]) that can be applied to some interpolation problems in d variables. First, an interpolation subproblem of degree n in d − 1 dimensions whose complexity Communicated by: Tomas Sauer Partially supported by the Spanish Research Grant MTM2012-31544 and by Gobierno de Arag´on and Fondo Social Europeo. J. M. Carnicer () Departamento de Matem´atica Aplicada/IUMA, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain e-mail: [email protected] C. God´es Departamento de Matem´atica Aplicada, Universidad de Zaragoza, Carretera de Cuarte s/n, 22071 Huesca, Spain
J.M. Carnicer, C. God´es
is lower has to be solved and then the problem can be reduced to a problem of degree n−1 in d dimensions. The recursion leads to suitable algorithms to solve the problem. When we choose an appropriate basis, this approach leads to the solution of a linear system whose coefficient matrix is lower triangular or block lower triangular in 2 dimensions. In the case of higher dimensions, the coefficient matrix has a hierarchical lower triangular structure which can be exploited to reduce considerably the number of computations to be done. The principal lattice (also called the Newton lattice) on a triangle is the set of points with barycentric coordinates (i/n, j/n, k/n), i + j + k ≤ n. Interpolation problems on principal lattices and related sets are widely used in the finite element method. Every principal lattice is a Berzolari-Radon set with lines parallel to the sides of the triangle. Generalized principal lattices [3–5] are also Berzolari-Radon sets and exhibit simple Lagrange and Newton formulae and generalizations of the Neville formula. Furthermore, a great variety of planar distributions of nodes with simple Lagrange and Newton formulae correspond to the Berzolari-Radon construction. The univariate Newton formula has not only practical applications but also some interest for the analysis of the interpolation problem offering new points of view. There exist different approaches to extend the univariate Newton formula. One of them consists of adding terms of degree n to an interpolant of degree n−1
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