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A qd-type method for computing generalized singular values of BF matrix pairs with sign regularity to high relative accuracy


Author: Rong Huang
Journal: Math. Comp. 89 (2020), 229-252
MSC (2010): Primary 65F15, 15A18, 15A23
DOI: https://doi.org/10.1090/mcom/3444
Published electronically: April 25, 2019
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Abstract: Structured matrices such as Vandermonde and Cauchy matrices frequently appear in various areas of modern computing, and they tend to be badly ill-conditioned, but a desirable property is that they admit accurate bidiagonal factorizations (BFs). In this paper, we propose a qd-type method to compute the generalized singular values of BF matrix pairs. A mechanism involving sign regularity of BF generators is provided to guarantee that there is no subtraction of like-signed numbers for the qd-type method. Consequently, all the generalized singular values are computed to high relative accuracy, independent of any conventional condition number. Error analysis and numerical experiments are presented to confirm the high relative accuracy.


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Additional Information

Rong Huang
Affiliation: School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, Hunan, People’s Republic of China
Email: rongh98@aliyun.com

DOI: https://doi.org/10.1090/mcom/3444
Keywords: Structured matrices, generalized singular values, qd-type methods, bidiagonal factorizations, high relative accuracy
Received by editor(s): May 24, 2018
Received by editor(s) in revised form: January 7, 2019, and February 25, 2019
Published electronically: April 25, 2019
Additional Notes: The author was supported by the National Natural Science Foundation of China (Grant Nos. 11871020 and 11471279) and the Natural Science Foundation for Distinguished Young Scholars of Hunan Province (Grant No. 2017JJ1025)
Article copyright: © Copyright 2019 American Mathematical Society