A Lagrange matrices approach to confluent Cauchy matrices
If \(L_{{\mathbf x},k}(x)\) are the Lagrange polynomials for the mutually distinct interpolation points \({\mathbf x}=(x_0,\ldots,x_n)\), and if \({\mathbf y}=(y_0,\ldots,y_n)\), is another such set of points then \([L_{{\mathbf x},k}(y_l)]\) is a Lagrange matrix. In the case of confluent interpolation points, the Lagrange polynomials are replaced by Sylvester polynomials and the Lagrange matrices by the corresponding Sylvester matrices. Also other polynomial bases such as Newton and Hermite polynomials, Vandermonde matrices, and Cauchy matrices associated with rational interpolation, and confluent forms thereof are all interconnected in this paper via the matrix relations. Explicit inversion formulas for the Lagrange and Cauchy matrices and for their confluent forms are given. When interpreted as Gram matrices for the polynomial bases and their dual evaluation functionals, then LU factorizations can be interpreted via a biorthogonalization process.
scientific article; zbMATH DE number 2117296