Accurate bidiagonal decomposition of totally positive Cauchy-Vandermonde matrices and applications

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• 作者：Ana Marco^a ; ^{ana.marco@uah.es} ; José ; -Javier Martí ; nez^a ; ^{jjavier.martinez@uah.es} ; Juan Manuel Peñ ; a^b ; ^{jmpena@unizar.es}
• 关键词：65F05 ; 65F15 ; 65D05 ; 15A23 ; 15B05 ; 15B48
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：517
• 期：Complete
• 页码：63-84
• 全文大小：437 K
• 卷排序：517

文摘

Cauchy–Vandermonde matrices play a fundamental role in rational interpolation theory and in other fields. When all their corresponding nodes are different and positive and all poles are different and negative and follow adequate orderings, these matrices are totally positive. In this paper we provide fast algorithms for computing bidiagonal factorizations of these matrices and their inverses with high relative accuracy. These algorithms can be used to solve with high relative accuracy other algebraic problems, such as the computation of all singular values, all eigenvalues or the solution of certain linear systems. The error analysis of the algorithm for computing the bidiagonal factorization and the corresponding perturbation theory are also performed.