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Accurate singular values of a class of parameterized negative matrices

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Abstract

Typically, parametrization captures the essence of a class of matrices, and its potential advantage is to make accurate computations possible. But, in general, parametrization suitable for accurate computations is not always easy to find. In this paper, we introduce a parametrization of a class of negative matrices to accurately solve the singular value problem. It is observed that, given a set of parameters, the associated nonsingular negative matrix can be orthogonally transformed into a totally nonnegative matrix in an implicit and subtraction-free way, which implies that such a set of parameters determines singular values of the associated negative matrix accurately. Based on this observation, a new O(n3) algorithm is designed to compute all the singular values, large and small, to high relative accuracy.

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Acknowledgements

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

Funding

The work of the first author was supported by the National Natural Science Foundation of China (Grant No. 11871020), 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). The work of the second author was supported by the National Science Foundation of China (Grant No. 11771100) and Laboratory of Mathematics for Nonlinear Science, Fudan University.

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Correspondence to Rong Huang.

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Communicated by: Raymond H. Chan

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Huang, R., Xue, J. Accurate singular values of a class of parameterized negative matrices. Adv Comput Math 47, 73 (2021). https://doi.org/10.1007/s10444-021-09898-z

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  • DOI: https://doi.org/10.1007/s10444-021-09898-z

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