A Lagrange matrices approach to confluent Cauchy matrices
type: Item
instance of: scholarly article
title: A Lagrange matrices approach to confluent Cauchy matrices
zbMATH Open document ID: 1063.65026
author: Luis Verde-Star
published in: Linear Algebra and its Applications
publication date: 2004-11-22T00:00:00Z
review text: 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.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 2117296
zbMATH Keywords: Cauchy matrices
zbMATH Keywords: matrix factorization
zbMATH Keywords: matrix inversion
zbMATH Keywords: Lagrange matrices
zbMATH Keywords: LU factorization
zbMATH Keywords: Sylvester matrices
zbMATH Keywords: Vandermonde matrices
zbMATH Keywords: inversion formulas
zbMATH Keywords: Gram matrices
MaRDI profile type: Publication
full work available at URL: j.laa.2004.03.018
OpenAlex ID: W2007907254
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DOI: 10.1016/J.LAA.2004.03.018
description: scientific article; zbMATH DE number 2117296