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research-article

Accurate singular values of a class of parameterized negative matrices

Published: 01 October 2021 Publication History

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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Cited By

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  • (2024)Computing eigenvalues of quasi-rational Said–Ball–Vandermonde matricesAdvances in Computational Mathematics10.1007/s10444-024-10191-y50:5Online publication date: 22-Aug-2024
  • (2024)Accurate computations of singular values and linear systems for Polynomial-Vandermonde-type matricesCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-024-00615-y61:4Online publication date: 24-Sep-2024
  • (2024)Accurate computations with some Catalan–Stieltjes matricesCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-024-00603-261:4Online publication date: 7-Oct-2024

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Published In

cover image Advances in Computational Mathematics
Advances in Computational Mathematics  Volume 47, Issue 5
Oct 2021
526 pages

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 October 2021
Accepted: 19 August 2021
Received: 14 April 2020

Author Tags

  1. Negative matrices
  2. Singular values
  3. High relative accuracy
  4. Parametrization

Author Tags

  1. 65F15
  2. 15A18
  3. 15B99

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Cited By

View all
  • (2024)Computing eigenvalues of quasi-rational Said–Ball–Vandermonde matricesAdvances in Computational Mathematics10.1007/s10444-024-10191-y50:5Online publication date: 22-Aug-2024
  • (2024)Accurate computations of singular values and linear systems for Polynomial-Vandermonde-type matricesCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-024-00615-y61:4Online publication date: 24-Sep-2024
  • (2024)Accurate computations with some Catalan–Stieltjes matricesCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-024-00603-261:4Online publication date: 7-Oct-2024

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