一类特殊Birkhoff插值问题适定插值基的计算

Computation of Proper Interpolation Basis for a Special Class of Birkhoff Interpolation Problems

摘要: 本文研究一类特殊的多元Birkhoff插值问题，插值条件由单个节点及其上的微分条件构成。首先针对节点为原点的情形，建立了关联矩阵与广义范德蒙矩阵之间的代数关系；其次对于一般节点，依据关联矩阵中相应列对应的单项构造上三角矩阵，进而得到广义范德蒙矩阵的计算方法，然后基于上述理论提出一种基于关联矩阵行约化的适定单项基的快速计算方法；最后用数值算例验证了方法的有效性与可行性。本文结果为多元Birkhoff插值适定性的判别与基构造提供了新的理论依据与计算工具。

Abstract: This paper investigates a class of special multivariate Birkhoff interpolation problems in which the interpolation conditions consist of a single node together with differential conditions imposed at that node. First, for the case where the node is located at the origin, the algebraic relationship between the incidence matrix and the generalized Vandermonde matrix is established. Furthermore, for general nodes, an upper triangular matrix is constructed from the monomials corresponding to the relevant columns of the incidence matrix, which yields a computational method for the generalized Vandermonde matrix. Then, based on the above theory, we propose a fast algorithm for computing the proper monomial basis via row reduction of the incidence matrix. Finally, numerical examples are provided to verify the effectiveness and feasibility of the proposed method. The results of this paper provide a new theoretical foundation and computational tool for the characterization of properness and the construction of basis in multivariate Birkhoff interpolation.