Accurate computations for eigenvalues of products of Cauchy-polynomial-Vandermonde matrices

Abstract

In this paper, we consider the product eigenvalue problem for the class of Cauchy-polynomial-Vandermonde (CPV) matrices arising in a rational interpolation problem. We present the explicit expressions of minors of CPV matrices. An algorithm is designed to accurately compute the bidiagonal decomposition for strictly totally positive CPV matrices and their additive inverses. We then illustrate the sign regularity of the bidiagonal decomposition to show that all the eigenvalues of a product involving such matrices are computed to high relative accuracy. Numerical experiments are given to confirm the claimed high relative accuracy.

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Acknowledgments

The authors would like to thank the Editor and the anonymous referees for their valuable comments and suggestions which have helped to improve the overall presentation of the paper.

Funding

This research was supported by the National Natural Science Foundations of China (Grants No. 11871020 and 11471279), the Natural Science Foundation for Distinguished Young Scholars of Hunan Province (Grant No. 2017JJ1025), and the Research Foundation of Education Bureau of Hunan Province (Grant No. 18A198).

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Correspondence to Rong Huang.

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Yang, Z., Huang, R. & Zhu, W. Accurate computations for eigenvalues of products of Cauchy-polynomial-Vandermonde matrices. Numer Algor 85, 329–351 (2020). https://doi.org/10.1007/s11075-019-00816-5

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Keywords

  • Product eigenvalue problems
  • Cauchy-polynomial-Vandermonde matrices
  • Bidiagonal decompositions
  • High relative accuracy